LOG MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE CURVES: THE FIRST FLIP

BRENDAN HASSETT AND DONGHOON HYEON

Abstract. We give a geometric invariant theory (GIT) construction of the log canonical model \overline{M}_{g}(α) of the pairs (\overline{M}_{g}, αδ) for α ∈ (7/10 − ϵ, 7/10] for small ϵ ∈ ℚ_{+}. We show that \overline{M}_{g}(7/10) is isomorphic to the GIT quotient of the Chow variety bicanonical curves; \overline{M}_{g}(7/10 − ϵ) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions, e.g., they do not contain elliptic bridges. We show that there is a small contraction Ψ : \overline{M}_{g}(7/10 + ϵ) ) \overline{M}_{g}(7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction Ψ^{+} : \overline{M}_{g}(7/10 − ϵ) ) \overline{M}_{g}(7/10) that is the Mori flip of Ψ.

Contents

1. Introduction 2 2. Statement of results and strategy of proof 5 3. GIT of Chow varieties and Hilbert schemes 14 4. Basin of attraction and equivalences 23 5. Computations over the moduli space of stable curves 26 6. Properties of c-semistable and h-semistable curves 30 7. Unstable bicanonical curves 36 8. Classification of curves with automorphisms 41 9. Interpreting the flip via GIT 46 10. Stability under one-parameter subgroups 49 11. Proofs of semistability and applications 61 References 73

Date: June 20, 2008.

2 BRENDAN HASSETT AND DONGHOON HYEON

1. Introduction

Our inspiration is to understand the canonical model of the moduli space \overline{M}_{g} of stable curves of genus g. This is known to be of general type for g = 22 and g ≥ 24 [Far06, HM82, EH86]. In these cases, we can consider the canonical ring

⊕_{n≥0}Γ(n(K_{\overline{M}_{g}} )).

which is finitely generated by a fundamental conjecture of birational geometry, recently proven in [BCHM06]. Then the corresponding projective variety

Proj ⊕_{n≥0} Γ(n(K_{\overline{M}_{g}} ))

is birational to \overline{M}_{g} and is called its canonical model.

There has been significant recent progress in understanding canonical models of moduli spaces. For moduli spaces A_{g} of principally polarized abelian varieties of dimension g ≥ 12, the canonical model exists and is equal to the first Voronoi compactification [SB06]. Unfortunately, no analogous results are known for \overline{M}_{g}, even for g ≫ 0.

Our approach is to approximate the canonical models with log canonical models. Consider α ∈ [0, 1] ∩ ℚ so that K_{\overline{M}_{g}} + αδ is an effective ℚ-divisor. We have the graded ring

⊕_{n≥0}Γ(n(K_{\overline{M}_{g}} + αδ))

and the resulting projective variety

\overline{M}_{g}(α) := Proj (⊕_{n≥0}Γ(n(K_{\overline{M}_{g}} + αδ))) .

Our previous paper [HHar] describes \overline{M}_{g}(α) explicitly for large values of α. For simplicity we assume that g ≥ 4: Small genera cases have been considered in [Has05, HL07b, HL07a]. For 9/11 < α ≤ 1, \overline{M}_{g}(α) is equal to \overline{M}_{g}. The first critical value is α = 9/11: \overline{M}_{g}(9/11) is the coarse moduli space of the moduli stack \overline{M}_{g}^{ps} of pseudostable curves [Sch91]. A pseudostable curve may have cusps but they are not allowed to have elliptic tails, i.e., genus one subcurves meeting the rest of the curve in one point. Furthermore, there is a divisorial contraction

T : \overline{M}_{g} ) \overline{M}_{g}(9/11)

induced by the morphism T : \overline{M}_{g} ) \overline{M}_{g}^{ps} of moduli stacks which replaces an elliptic tail with a cusp. Furthermore, \overline{M}_{g}(α) ≃ \overline{M}_{g}(9/11) provided 7/10 < α ≤ 9/11.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 3

This paper addresses what happens when α = 7/10. Given a sufficiently small positive ϵ ∈ ℚ, we construct a small contraction and its flip:

\xymatrix{\overline{M}_{g}(\frac{7}{10} + ϵ) \ar@{-->}[dr]^{Ψ} &  & \overline{M}_{g}(\frac{7}{10} − ϵ) \ar@{-->}[dl]_{Ψ^{+}} \\  & \overline{M}_{g}(\frac{7}{10})}

The resulting spaces arise naturally as geometric invariant theory (GIT) quotients and admit partial modular descriptions. We construct \overline{M}_{g}(7/10) as a GIT quotient of the Chow variety of bicanonical curves; it parametrizes equivalence classes of c-semistable curves. We defer the formal definition, but these have nodes, cusps, and tacnodes as singularities. The flip \overline{M}_{g}(7/10 − ϵ) is a GIT quotient of the Hilbert scheme of bicanonical curves; it parametrizes equivalence classes of h-semistable curves, which are c-semistable curves not admitting certain subcurves composed of elliptic curves (see Definition 2.6).

Figure 1. Geometry of the flip

We may express the flip in geometric terms (Figure 1): Let C = D ∪_{p,q} E denote an elliptic bridge, where D is smooth of genus g − 2, E is smooth of genus one, and D meets E at two nodes p and q. Let C′ be a tacnodal curve of genus g, with normalization D and conductor \{p, q\}. In passing from \overline{M}_{g}(7/10 + ϵ) to \overline{M}_{g}(7/10 − ϵ), we replace C with C′. Note that the descent data for C′ includes the choice of an isomorphism of tangent spaces

ι : T_{p}D −)^{∼} T_{q}D;

4 BRENDAN HASSETT AND DONGHOON HYEON

the collection of such identifications is a principal homogeneous space for 𝔾_{m}. When C is a generic elliptic bridge, the fiber (Ψ^{+})^{−1}(Ψ(C)) ≃ ℙ^{1}; see Proposition 6.5 for an explicit interpretation of the endpoints.

Here we offer a brief summary of the contents of this paper; a more detailed roadmap can be found in Section 2.5. Section 3 is devoted to a general discussion of the GIT of Chow points and Hilbert points. The main applications are the analysis of tautological classes and polarizations on the Hilbert scheme, the resulting formulas for Hilbert-Mumford indices (Proposition 3.17), and cycle maps (Corollary 3.14). We also recall various formulations of the Hilbert-Mumford one-parameter-subgroup criterion.

Section 4 is a brief review of the basin-of-attraction techniques used in this paper. These are important for analyzing when distinct curves are identified in the GIT quotients.

Section 5 discusses, in general terms, how to obtain contractions of the moduli space of stable curves from GIT quotients of Hilbert schemes. The resulting models of the moduli space depend on the choice of linearization; we express the polarizations in terms of tautological classes.

Section 6 summarizes basic properties of c-semistable curves: embedding theorems and descent results for tacnodal curves. Section 7 offers a preliminary analysis of the GIT of the Hilbert scheme and the Chow variety of bicanonically embedded curves of genus g ≥ 4. Then in Section 8 we enumerate the curves with positive-dimensional automorphism groups. Section 9 applies this to give a GIT construction of the flip Ψ^{+} : \overline{M}_{g}(7/10 − ϵ) ) \overline{M}_{g}(7/10).

Section 10 offers a detailed orbit closure analysis, using basins of attractions and a careful analysis of the action of the automorphism group on tangent spaces. The main application is a precise description of the semistable and stable bicanonical curves, proven in Section 11.

Throughout, we work over an algebraically closed field k, generally of characteristic zero. However, Sections 3 and 6 are valid in positive characteristic.

Acknowledgments: The first author was partially supported by National Science Foundation grants 0196187, 0134259, and 0554491, the Sloan Foundation, and the Institute of Mathematical Sciences of the Chinese University of Hong Kong. The second author was partially supported by the Korea Institute for Advanced Study and the Korea Science and Engineering Foundation (KOSEF)

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 5

grant funded by the Korea government (MOST) (No. R01-2007-000-10948-0). We owe a great deal to S. Keel, who helped shape our understanding of the birational geometry of \overline{M}_{g} through detailed correspondence. We are also grateful to D. Abramovich, Y. Kawamata, I. Morrison, B.P. Purnaprajna, M. Simpson, D. Smyth, and D. Swinarski for useful conversations.

2. Statement of results and strategy of proof

2.1. Stability notions for algebraic curves. In this paper, we will use four stability conditions: Deligne-Mumford stability [DM69], Schubert pseudostability [Sch91], c-(semi)stability, and h-(semi)stability. We recall the definition of pseudostability, which is obtained from Deligne-Mumford stability by allowing ordinary cusps and prohibiting elliptic tails:

Definition 2.1. [Sch91] A complete curve is pseudostable if

(1) it is connected, reduced, and has only nodes and ordinary cusps as singularities;

(2) admits no elliptic tails, i.e., connected subcurves of arithmetic genus one meeting the rest of the curve in one node;

(3) the canonical sheaf of the curve is ample. The last condition means that each subcurve of genus zero meets the rest of the curve in at least three points.

Before formulating the notions of c- and h-(semi)stability, we need the following definition:

Definition 2.2. An elliptic bridge is a connected subcurve of arithmetic genus one meeting the rest of the curve in two nodes.

In our stability analysis, we will require additional technical definitions:

Definition 2.3. An open elliptic chain of length r is a two-pointed projective curve (C′, p, q) such that

• C′ = E_{1} ∪_{a_{1}} ⋯ ∪_{a_{r−1}} E_{r} where each E_{i} is connected of genus one, with nodes, cusps or tacnodes as singularities;

• E_{i} intersects E_{i+1} at a single tacnode a_{i}, for i = 1, …, r − 1;

• E_{i} ∩ E_{j} = ∅ if |i − j| > 1;

• p, q ∈ C′ are smooth points with p ∈ E_{1} and q ∈ E_{r};

Figure 2. Generic elliptic bridges

• ω_{C′}(p + q) is ample.

An open elliptic chain of length r has arithmetic genus 2r − 1.

Figure 3. Generic elliptic chain of length three

Definition 2.4. Let C be a projective connected curve of arithmetic genus g ≥ 3, with nodes, cusps, and tacnodes as singularities. We say C admits an open elliptic chain if there is an open elliptic chain (C′, p, q) and a morphism ι : C′ ) C such that

• ι is an isomorphism over C′ \ {p, q} onto its image;

• ι(p), ι(q) are nodes of C; we allow the case ι(p) = ι(q), in which case C

is said to be a closed elliptic chain. C admits a weak elliptic chain if there exists ι : C′ ) C as above with the second condition replaced by

• ι(p) is a tacnode of C and ι(q) is a node of C; or

• ι(p) = ι(q) is a tacnode of C, in which case C is said to be a closed weak elliptic chain.

Now we are in a position to formulate our main stability notions:

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 7

Figure 4. Generic weak elliptic chain

Definition 2.5. A complete curve C is said to be c-semistable if (1) C has nodes, cusps and tacnodes as singularities;

(2) ω_{C} is ample;

(3) a connected genus one subcurve meets the rest of the curve in at least

two points (not counting multiplicity). It is said to be c-stable if it is c-semistable and has no tacnodes or elliptic bridges.

Definition 2.6. A complete curve C of genus g is said to be h-semistable if it is c-semistable and admits no elliptic chains. It is said to be h-stable if it is h-semistable and admits no weak elliptic chains.

Remark 2.7. A curve is c-stable if and only if it is pseudostable and has no elliptic bridges.

Table 1 summarizes the defining characteristics of the stability notions.

2.2. Construction of the small contraction Ψ. We start with some preliminary results. Recall from [HHar] that T : \overline{M}_{g} ) \overline{M}_{g}^{ps} denotes the functorial contraction and T : \overline{M}_{g} ) \overline{M}_{g}^{ps} = \overline{M}_{g}(9/11) the induced morphism on coarse moduli spaces, which contracts the divisor Δ_{1}.

Lemma 2.8. For α < 9/11, (\overline{M}_{g}^{ps}, αδ^{ps}) and (\overline{M}_{g}^{ps}, αΔ^{ps}) are log terminal and

\overline{M}_{g}(α) ≃ Proj (⊕_{n≥0}Γ(n(K\overline{M} _{g}^{ps} + αδ^{ps}))) .

Proof. Since g > 3, the locus in \overline{M}_{g}^{ps} parametrizing curves with nontrivial automorphisms has codimension ≥ 2 [HM82, §2]. (Of course, we have already collapsed δ_{1}.) Thus the coarse moduli map q : \overline{M}_{g}^{ps} ) \overline{M}_{g}^{ps} is unramified in codimension one and

(2.1)  q^{∗}(K\overline{M} _{g}^{ps} + αΔ^{ps}) = K\overline{M}_{g}^{ps} + αδ^{ps}

Table 1. Stability notions

singularity genus zero subcurve meets the rest in ... genus one subcurve meets the rest in ... elliptic chain weak elliptic chain

stable  nodes  ≥ 3 points  –  –  –
pseudostable  nodes, cusps  ≥ 3 points  ≥ 2 points  –  –

c-semistable nodes, cusps, tacnodes ≥ 3 points counting multipilicity

≥ 2 points  –  –

c-stable  nodes, cusps  ≥ 3 points  ≥ 3 points  –  –

h-semistable nodes, cusps, tacnodes ≥ 3 points counting multipilicity ≥ 3 points counting multipilicity not admitted

–

h-stable nodes cusps, tacnodes ≥ 3 points counting multipilicity ≥ 3 points counting multipilicity not admitted

not admitted

for each α. We have the log discrepancy equation [HHar, §4]

(2.2)  K_{\overline{M}_{g}} + αδ = T^{∗}(K\overline{M}_{g}^{ps} + αδ^{ps}) + (9 − 11α)δ_{1}

and the pull back

T^{∗}(K\overline{M} _{g}^{ps} + 7/10δ^{ps}) = K_{\overline{M}_{g}} + 7/10δ − 13/10δ_{1} ∼ 10λ − δ − δ_{1},

where ∼ designates proportionality.

Since \overline{M}_{g} is smooth and δ is normal crossings, the pair (\overline{M}_{g}, αδ+(11α−9)δ_{1}) is log terminal. The discrepancy equation implies that (\overline{M}_{g}^{ps}, αδ^{ps}) is log terminal for α ∈ [7/10, 9/11). Applying the ramification formula [KM98, 5.20] to (2.1) (or simply applying [HHar, A.13]), we find that (\overline{M}_{g}^{ps}, αΔ^{ps}) is also log terminal.

Since Δ_{1} is T-exceptional, for each Cartier divisor L on \overline{M}_{g}^{ps} and m ≥ 0 we have Γ(\overline{M}_{g}, T^{∗}L + mΔ_{1}) ≃ Γ(\overline{M}_{g}^{ps}, L). This implies that

\overline{M}_{g}(α) = Proj ⊕_{n≥0} Γ(\overline{M}_{g}, n(K_{\overline{M}_{g}} + αδ))
= Proj ⊕_{n≥0} Γ (\overline{M}_{g}, n(T^{∗}(K\overline{M} _{g}^{ps} + αδ^{ps}) + (9 − 11α)δ_{1}))
≃ Proj ⊕_{n≥0} Γ (\overline{M}_{g}^{ps}, n(K\overline{M} _{g}^{ps} + αδ^{ps})) .
□

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 9

We shall construct the contractions by using the powerful results of [GKM02]:

Proposition 2.9. For α ∈ (7/10, 9/11] ∩ ℚ, there there exists a birational contraction

Ψ : \overline{M}_{g}(α) ) \overline{M}_{g}(7/10).

It contracts the codimension-two strata T_{i}, i = 0, 2, …, ⌊(g − 1)/2⌋, where (1) T_{0} = {E ∪_{p,q} D | g(E) = 1, g(D) = g − 2};

(2) T_{i} = {C_{1} ∪_{p} E ∪_{q} C_{2} | g(C_{1}) = i, g(E) = 1, g(C_{2}) = g − 1 − i}, 2 ≤ i ≤

⌊(g − 1)/2⌋, by collapsing the loci \overline{M}_{1,2} ⊂ T_{i} corresponding to varying (E, p, q).

Remark 2.10. We shall see in Corollary 2.15 that Ψ is an isomorphism away from T_{•} := ∪T_{i}.

Proof. Recall that K\overline{M} _{g}^{ps} + αΔ^{ps} is ample provided 7/10 < α ≤ 9/11; this is part of the assertion that \overline{M}_{g}(α) = \overline{M}_{g}^{ps} for 7/10 < α ≤ 9/11 [HHar, Theorem 1.2]. However, K\overline{M}_{g}^{ps} + 7/10Δ^{ps} is nef but not ample [HHar, §4]. Indeed, the pull-back to \overline{M}_{g}

10λ − δ − δ_{1}

can be analyzed using the classification of one-dimensional boundary strata by Faber [Fab96] and Gibney-Keel-Morrison [GKM02]. It is ‘F-nef’, in the sense that it intersects all these strata nonnegatively, and is therefore nef by [GKM02, 6.1]. Later on, we will list the strata meeting it with degree zero.

We apply Kawamata basepoint freeness [KM98, 3.3]:

Let (X, D) be a proper Kawamata log terminal pair with D effective. Let M be a nef Cartier divisor such that aM − K_{X} − D is nef and big for some a > 0. Then |bM| has no basepoint for all b ≫ 0.

For our application, M is a Cartier multiple of K\overline{M} _{g}^{ps} + 7/10Δ^{ps} and D = (7/10 − ϵ)Δ^{ps} for small positive ϵ ∈ ℚ. The resulting morphism is denoted Ψ.

We claim that Ψ is birational. To establish the birationality, we show that each curve B ⊂ \overline{M}_{g} meeting the interior satisfies

B.(10λ − δ − δ_{1}) > 0.

The Moriwaki divisor

A := (8g + 4)λ − gδ_{0} − ∑_{i=1}^{⌊g/2⌋} 4i(g − i)δ_{i}

meets each such curve nonnegatively [Mor98, Theorem B]. We can write

10λ − δ − δ_{1} = (1/g) A + (2 − 4/g)λ + (2 − 4/g)δ_{1} + ∑_{i=2}^{⌊g/2⌋}(−1 + 4i(g − i)/g)δ_{i}.

Each of these coefficients is positive: Clearly 1/g, 2−4/g > 0 and since 2i/g ≤ 1,

−1 + 4i(g − i)/g = −1 + 4i − (2i/g)2i ≥ −1 + 4i − 2i > 0.

Thus we have

B.(10λ − δ − δ_{1}) ≥ (2 − 4/g)λ.B > 0,

where the last inequality reflects the fact that the Torelli morphism is nonconstant along B.

We verify the image of Ψ equals \overline{M}_{g}(7/10). The log discrepancy formula (2.2) implies

Image(Ψ) = Proj ⊕_{n≥0} Γ(n(K_{\overline{M}_{g}} + 7/10δ − 13/10δ_{1})).

However, since Δ_{1} is (Ψ ∘ T)-exceptional adding it does not change the space of global sections, whence

Image(Ψ) = Proj ⊕_{n≥0} Γ(n(K_{\overline{M}_{g}} + 7/10δ)) = \overline{M}_{g}(7/10).

Finally, we offer a preliminary analysis of the locus contracted by Ψ. The main ingredient is the enumeration of one-dimensional boundary strata in [GKM02] (see also [HHar, §4]). We list the ones orthogonal to 10λ − δ − δ_{1}; any stratum swept out by these classes is necessarily contracted by Ψ. In the second and third cases X_{0} denotes a varying 4-pointed curve of genus zero parametrizing the stratum.

(1) Families of elliptic tails, which sweep out δ_{1} and correspond to the extremal ray contracted by T .

(2) Attach a 2-pointed curve of genus 0 and a 2-pointed curve (D, p, q) of genus g − 2 to X_{0} and stabilize. Contracting this and the elliptic tail stratum collapses T_{0} along the \overline{M}_{1,2}’s corresponding to fixing (D, p, q) and varying the other components.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 11

(3) Attach a 1-pointed curve (C_{1}, p) of genus i > 1, a 1-pointed curve (C_{2}, q) of genus g − 1 − i > 1, and a 2-pointed curve of genus 0 to X_{0} and stabilize. Contracting this and the elliptic tail stratum collapses T_{i} along the \overline{M}_{1,2}’s corresponding to fixing (C_{1}, p), (C_{2}, q) and varying the other components.

Thus the codimension-two strata T_{0}, T_{2}, …, T_{⌊(g−1)/2⌋} are all contracted by Ψ. □

2.3. Construction of the flip Ψ^{+}. Consider the Chow variety of degree 4g − 4 curves of genus g in ℙ^{3g−4}. Let Chow_{g,2} denote the closure of the bicanonically embedded smooth curves of genus g. Similarly, let Hilb_{g,2} denote the closure of these curves in the Hilbert scheme.

Proposition 2.11. The cycle class map

(2.3)  ϖ : Hilb_{g,2} ) Chow_{g,2}

induces a morphism of GIT quotients

Hilb_{g,2}^{ss}//SL_{3g−3} ) Chow_{g,2}^{ss}//SL_{3g−3},

where the Hilbert scheme has the asymptotic linearization introduced in §3.6.

This is a special case of Corollary 3.14, which applies quite generally to cycleclass maps from Hilbert schemes for Chow varieties. (See §3.6 for background information on the cycle class map.) Let \overline{M}_{g}^{hs} and \overline{M}_{g}^{cs} denote the resulting GIT quotients Hilb_{g,2}^{ss}//SL_{3g−3} and Chow_{g,2}^{ss}//SL_{3g−3}, and

(2.4)  Ψ^{+} : \overline{M}_{g}^{hs} ) \overline{M}_{g}^{cs}

the morphism of Proposition 2.11.

Theorem 2.12. Let ϵ ∈ ℚ be a small positive number. There exist isomorphisms

(2.5)  \overline{M}_{g}(7/10) ≃ \overline{M}_{g}^{cs}

and

(2.6)  \overline{M}_{g}(7/10 − ϵ) ≃ \overline{M}_{g}^{hs}

such that the induced morphism

Ψ^{+} : \overline{M}_{g}(7/10 − ϵ) ) \overline{M}_{g}(7/10)

is the flip of Ψ.

We thus obtain a modular/GIT interpretation of the flip:

\xymatrix{\overline{M}_{g}(\frac{7}{10} + ϵ) ≃ \overline{M}_{g}^{ps} \ar@{-->}[dr]^{Ψ} &  & \overline{M}_{g}(\frac{7}{10} − ϵ) ≃ \overline{M}_{g}^{hs} \ar@{-->}[dl]_{Ψ^{+}} \\  & \overline{M}_{g}(\frac{7}{10}) ≃ \overline{M}_{g}^{cs}}

2.4. Stability results on bicanonical curves. For c-semistable curves, ω_{C}^{⊗2} is very ample and has no higher cohomology (Proposition 6.1). The image in ℙ^{3g−4} is said to be bicanonically embedded.

Theorem 2.13. The semistable locus Chow_{g,2}^{ss} (resp. stable locus Chow_{g,2}^{s}) corresponds to bicanonically embedded c-semistable (resp. c-stable) curves.

Unlike in \overline{M}_{g} and \overline{M}_{g}^{ps}, nonisomorphic curves may be identified in the quotient Chow_{g,2}//SL_{3g−3}. For example, if a c-semistable curve C = D ∪_{p,q} E consists of a genus g − 2 curve D meeting in two nodes p, q with an elliptic curve E, then it is identified with any tacnodal curve obtained by replacing E with a tacnode. In Section 10, we shall give a complete classification of strictly semistable curves and the curves in their orbit closure.

Theorem 2.14. The semistable locus Hilb_{g,2}^{ss} (resp. stable locus Hilb_{g,2}^{s}) with respect to the asymptotic linearization corresponds to bicanonically embedded hsemistable (resp. h-stable) curves.

One difference from the case of Chow points is that tacnodal curves may well be Hilbert stable. For instance, when g ≥ 4 irreducible bicanonical h-semistable curves are necessarily h-stable. When g = 3, a bicanonical h-semistable curve is Hilbert strictly semistable if and only if it has a tacnode [HL07a]. When g = 4, every h-semistable curve is h-stable and the moduli functor is thus separated.

Since c-stable curves are h-stable (see Proposition 3.13) and pseudostable (see Remark 2.7), we have

Corollary 2.15. Ψ and Ψ^{+} are isomorphisms over the locus of c-stable curves. Thus Ψ is a small contraction with exceptional locus T_{•} and Ψ^{+} is a small contraction with exceptional locus Tac, the h-semistable curves with tacnodes.

Thus the geometry of the flip is as indicated in Figure 1: Ψ^{+}(C′) = Ψ(C) precisely when C is the ‘pseudostable reduction’ of C′.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 13

2.5. Detailed roadmap for the GIT analysis. The proof of Theorems 2.13 and 2.14 is rather intricate, so we give a bird’s eye view for the reader’s convenience.

(1) The following implications are straightforward:

• From the definitions, it is clear that:

h-semistable } c-semistable

• General results on linearizations of Chow and Hilbert schemes (Proposition 3.13) imply

Hilbert semistable } Chow semistable

and Chow stable } Hilbert stable.

(2) We next prove that non c-semistable (resp. non h-semistable) curves are Chow unstable (resp. Hilbert unstable). The main tool is the stability algorithm (Proposition 3.7).

• Non c-semistable curves can be easily destabilized by one-parameter subgroups (§7). We obtain

Chow semistable } c-semistable .

• We show that if a curve C admits an open rosary of even length (see Definition 8.1), then there is a 1-PS ρ coming from the automorphism group of the rosary such that the mth Hilbert point [C]_{m} (Definition 3.5) is unstable with respect to ρ for all m ≥ 2 (Proposition 10.1 and Proposition 10.7).

• If C admits an elliptic chain, then it is contained in the basin of attraction A_{ρ}([C_{0}]_{m}) (see Definition 4.1) of a curve C_{0} admitting an open rosary of even length such that μ([C_{0}]_{m}, ρ) < 0. Hence such curves are Hilbert unstable (Propositions 10.3 and 10.8) and we obtain

Hilbert semistable } h-semistable .

(3) We prove “c-semistable } Chow semistability”, and use it to establish “hsemistability } Hilbert semistability”.

14 BRENDAN HASSETT AND DONGHOON HYEON

• The only possible Chow semistable replacement of a c-stable curve is itself (see Theorem 9.1). Thus c-stable curves are Chow stable and hence Hilbert stable.

• We show that any strictly c-semistable curve C is contained in a basin of attraction of a distinguished c-semistable curve C^{⋆} with one-parameter isomorphism such that μ(Ch(C^{⋆}), ρ) = 0 (see Proposition 11.5). Indeed, we choose C^{⋆} so that it has closed orbit in the locus of c-semistable points (cf. Proposition 11.6).

• If C is strictly c-semistable, its pseudo-stabilization D has elliptic bridges. For any such D, there is a distinguished strictly c-semistable curve C^{⋆} such that its basins of attraction contain every c-semistable replacement for D. Futhermore, every possible Chow-semistable replacement for D is contained in some basin of attraction A_{ρ′}(Ch(C^{⋆})) with μ(Ch(C^{⋆}), ρ′) = 0. Since one of these must be Chow semistable, every one is Chow semistable (see Lemma 4.3).

• The Hilbert semistable curves form a subset of the set of Chow semistable curves. We first identify the Chow semistable curves admitting oneparameter subgroups that are Hilbert-destabilizing. Then we show that any curve that is Hilbert unstable but Chow semistable arises in the basin of attraction of such a curve. These basins of attraction consist of the curves that are c-semistable but not h-semistable. Thus the h-semistable curves are Hilbert semistable (§11.3).

3. GIT of Chow varieties and Hilbert schemes

Let ℙ^{N} = ℙ(V) for some (N + 1)-dimensional vector space V. Throughout this section, let ρ : 𝔾_{m} ) GL(V) be a one-parameter subgroup and x_{0}, …, x_{N} be homogeneous coordinates that diagonalize the ρ-action so that

ρ(t).x_{i} = t^{r_{i}} x_{i}, i = 0, …, N, r_{0} ≥ ⋯ ≥ r_{N} = 0.

We have the associated one-parameter subgroup ρ^{∘} : 𝔾_{m} ) SL(V)

ρ^{∘}(t).x_{i} = t^{r_{i} −(r_{0}+...+r_{N} )/(N+1)} x_{i}.

Given x ∈ ℙ(V), the Hilbert-Mumford index is given by (cf. [MFK94, 2.1]):

μ(x, ρ) = max{−r_{i} + (r_{0} + … + r_{N})/(N + 1) : x_{i} ≠ 0}.

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We say that x is (semi)stable with respect to ρ^{∘} if μ(x, ρ) > (≥) 0. A fundamental theorem of GIT is that x is GIT (semi)stable if and only if it is (semi)stable with respect to every 1-PS of SL(V). We will sometimes abuse terminology and say that x is (semi)stable with respect to ρ when it is (semi)stable with respect to ρ^{∘}.

3.1. GIT of Chow points. We briefly recall Mumford’s interpretation of Hilbert-Mumford criterion for Chow stability of projective varieties [Mum77]. Let X ⊂ ℙ(V) be a projective variety and ρ be a one-parameter subgroup of GL(V) with weights r_{0} ≥ r_{1} ≥ ⋯ ≥ r_{N} = 0. Let I_{ρ} be the ideal sheaf of O_{X}[t] such that

I_{ρ} · O_{X}(1)[t] = ( \begin{matrix}the O_{X}-submodule of O_{X}(1)[t] generated by \\ t^{r_{i}} x_{i}, i = 0, …, N. \end{matrix} )

Definition 3.1. The Hilbert-Samuel multiplicity e_{ρ}(X) is the normalized leading coefficient of P(n) := χ(L^{n}/I_{ρ}^{n}L^{n}) where L is the invertible O_{X}[t]-module O_{X}(1)[t].

Then the Hilbert-Mumford criterion can be translated in terms of e_{ρ}(X) as follows:

Theorem 3.2. [Mum77, Theorem 2.9] The Chow point of X is stable (resp. semistable) if and only if

e_{ρ}(X) < (resp. ≤)\frac{dim(X) + 1}{N + 1}deg (X) ∑ r_{i}

for any one-parameter subgroup ρ : 𝔾_{m} ) GL(V) with weights r_{0} ≥ r_{1} ≥ ⋯ ≥ r_{N} = 0.

We shall make frequent use of the following lemma which describes how the Hilbert-Samuel multiplicity is affected by the singular points:

Lemma 3.3. [Sch91, Lemma 1.4] Let X be a reduced curve in ℙ(V) and ν : X̃) X be its normalization.

(1) e_{ρ}(X) = ∑ _{P∈X̃}e_{ρ}(X̃)_{P}, where e_{ρ}(X̃)_{P} denotes the normalized leading coefficient of dim_{k} O_{X̃×𝔸^{1}}/I_{P×{0}}^{m}.

(2) Suppose that v(ν^{∗}x_{i}) + r_{i} ≥ a for all i where v is the natural valuation of O_{P,X̃}. Then e_{ρ}(X̃) ≥ a^{2}.

We shall also use the following lemma that addresses the case in which X is degenerate and ρ acts on X trivially.

Lemma 3.4. [Sch91, Lemma 1.2] Let X be an r-dimensional variety in ℙ^{N}. Let ρ be a 1-PS of GL_{N+1}(k) such that ρ(t) · x_{i} = t^{r_{i}} x_{i}, r_{0} ≥ ⋯ ≥ r_{N} = 0. Suppose that x_{j}, x_{j+1}, …, x_{N} vanish on X and r_{0} = r_{1} = ⋯ = r_{j−1}. Then

e_{ρ}(X) = (r + 1)r_{0}deg (X).

3.2. GIT of Hilbert points. Let X ⊂ ℙ^{N} = ℙ(V) be a projective variety with Hilbert polynomial P(t). Choose an integer m sufficiently large so that

• O_{X}(m) has no higher cohomology;

• the natural map

Sym^{m}V^{∗} ) Γ(O_{X}(m))

is surjective.

Definition 3.5. The mth Hilbert point [X]_{m} of X is defined

[X]_{m} := [Sym^{m}V^{∗} ) Γ(O_{X}(m))] ∈ Gr(P(m), Sym^{m}V) ) ℙ( ⋀^{P(m)} Sym^{m}V).

Note that X is determined by [X]_{m} provided X is cut out by forms of degree m.

Definition 3.6. X is said to be m-Hilbert stable (resp. semistable) if [X]_{m} is GIT stable (semistable) with respect to the natural SL(V) action on ℙ(⋀^{P(m)} Sym^{m}V).

We refer the reader to [HHL07] for detailed discussion of an algorithm (and a Macaulay 2 implementation) using Gröbner basis to determine whether a variety is m-Hilbert (semi)stable with respect to a given one-parameter subgroup. We sketch the main results here.

For any given v ∈ ℝ^{N+1}, ≺_{v} denotes the monomial order defined by declaring x^{a} ≺_{v} x^{b} if

(1) deg x^{a} < deg x^{b};

(2) deg x^{a} = deg x^{b} and v.a < v.b;

(3) deg x^{a} = deg x^{b}, v.a = v.b and x^{a} ≺_{Lex} x^{b} in the lexicographic order. In particular, given a one-parameter subgroup ρ with the weight vector w = (r_{0}, …, r_{N}), the monomial order ≺_{ρ} means the graded lexicographic order associated to the weight w. Given a monomial x^{a} = x_{0}^{a_{0}} ⋯ x_{N}^{a_{N}} , the ρ-weight is

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 17

defined

wt_{ρ}(x^{a}) := w.a = r_{0}a_{0} + … + r_{N}a_{N}.

For each polynomial f, let in_{≺_{ρ}}(f) denote the largest term of f with respect to ≺_{ρ}. For an ideal I ⊂ SymV^{∗}, we let in_{≺_{ρ}}(I) := ⟨in_{≺_{ρ}}(f) | f ∈ I⟩. Let I ⊂ SymV^{∗} be a homogeneous ideal with graded pieces I_{m} = I ∩ Sym^{m}V^{∗}. The monomials {x^{a(1)}, …, x^{a(P(m))} } of degree m not contained in in_{≺_{ρ}}(I) form a basis for Sym^{m}V^{∗}/I_{m}.

We reformulate Gieseker’s stability criterion for Hilbert points [Gie82, pp. 8] in these terms:

Proposition 3.7. The Hilbert-Mumford index of [X]_{m} with respect to a oneparameter subgroup ρ : 𝔾_{m} ) GL(V) with weights r_{0}, r_{1}, …, r_{N} is given by

(3.1)  μ([X]_{m}, ρ) = \frac{mP(m)}{N + 1} ∑ r_{i} − ∑_{j=1}^{P(m)} wt_{ρ}(x^{a(j)} )

where a(1), …, a(P(m)) index the monomials of degree m not contained in in_{≺_{ρ}}(I). In particular, [X]_{m} ∈ ℙ(⋀^{P(m)} Sym^{m}V) is stable (resp. semistable) under the natural SL(V)-action if and only if for any one-parameter subgroup ρ we have

∑_{j=1}^{P(m)} wt_{ρ}(x^{a(j)} ) < (resp. ≤)\frac{mP(m)}{N + 1} ∑ r_{i}.

3.3. Polarizations on Hilbert schemes. Let Hilb be the connected component of the Hilbert scheme containing X, X ⊂ ℙ(V) × Hilb the universal family, π : X ) Hilb the natural projection, and O_{X} (1) the polarization.

A coherent sheaf F on ℙ(V) is said to be M-regular in the sense of Castelnuovo and Mumford [Mum66, ch. 14] if H^{i}(F(M−i)) = 0 for each i > 0. Suppose that the ideal sheaf I_{X} is M-regular. It follows that for each m ≥ M

• Γ(I_{X}(m)) ⊗ V^{∗} ) Γ(I_{X}(m + 1)) is surjective;

• H^{i}(I_{X}(m − i)) = 0 for each i > 0;

and also • Sym^{m−1}V^{∗} ) Γ(O_{X}(m − 1)) is surjective;

• H^{i}(O_{X}(m − 1 − i)) = 0 for i > 0.

In particular, O_{X} is (M − 1)-regular. Conversely, if O_{X} is (M − 1)-regular and M ≥ 0 then I_{X} is M-regular. [Eis05, pp. 68]

18 BRENDAN HASSETT AND DONGHOON HYEON

There exists an M ≫ 0 such that every [X] ∈ Hilb has M-regular ideal sheaf [Mum66, ch.14]. Then for each m ≥ M we get a closed embedding [Mum66, ch. 15]

Hilb ⊂ Gr(P(m), Sym^{m}V) ⊂ ℙ(⋀^{P(m)} Sym^{m}V)
[X] ) [X]_{m}

The universal quotient bundle Q ) Gr(P(m), Sym^{m}V) satisfies

Q|Hilb = π_{∗}O_{X} (m),

and on taking determinants we find

Λ_{m} := O_{Gr}(1)|Hilb = det(π_{∗}O_{X} (m)).

3.4. Tautological classes on the Hilbert scheme. Recall the tautological divisor classes developed in [Fog69] and [KM76, Theorem 4]: There exist Cartier divisors L_{0}, …, L_{r+1} on Hilb such that

(3.2)  det(ℝ^{•}π_{∗}O_{X} (m)) = ∑_{i=0}^{r+1} \binom{m}{i}L_{i},

where r is the dimension of subschemes parametrized by Hilb. That is, the determinant of cohomology of O_{X} (m) can be expressed as a polynomial in the tautological class. This is a relative version of the Hilbert polynomial of X over Hilb. It follows that the polarizations introduced above satisfy:

(3.3)  Λ_{m} = det(π_{∗}O_{X} (m)) = ∑_{i=0}^{r+1} \binom{m}{i}L_{i}.

Using these formulas, we extend our definition:

Definition 3.8. For each m ∈ ℤ, write

Λ_{m} = det(ℝ^{•}π_{∗}O_{X} (m)) = ∑_{i=0}^{r+1} \binom{m}{i}L_{i}.

In many situations the tautological divisors satisfy a dependence relation:

Proposition 3.9. Let Hilb denote a connected component of the Hilbert scheme parametrizing subschemes in ℙ(V) of dimension r and L_{0}, …, L_{r+1} the tautological divisors on Hilb. Let Hilb^{•,1} ⊂ Hilb denote an open subset corresponding to subschemes X where the following hold:

• O_{X}(1) has no higher cohomology;

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 19

• the restriction map V^{∗} ) Γ(O_{X}(1)) is an isomorphism.

Over Hilb^{•,1} we have the relation L_{0} + L_{1} = 0.

In particular, if r = 1 then Equation 3.3 takes the form

(3.4)  Λ_{m} = L_{0} + mL_{1} + \frac{m(m − 1)}{2} L_{2} = (m − 1)(L_{1} + \frac{m}{2}L_{2}).

Proof. Let π : X ) Hilb^{•,1} be the universal family embedded in ℙ(V) × Hilb^{•,1}. Our first assumption implies π_{∗}O_{X} (1) is locally free and

Λ_{1} = det(π_{∗}O_{X} (1)) = L_{0} + L_{1}.

The second assumption implies we have a trivialization (cf. [Vie95, pp.44])

Γ(O_{ℙ(V)}(1)) ⊗ O_{Hilb^{•,1}} ≃ π_{∗}O_{X} (1).

In particular, it follows that L_{0} + L_{1} = 0. □

3.5. Hilbert points and Hilbert schemes. We have seen that Hilb admits an embedding into Gr(P(m), Sym^{m}V) for m ≫ 0. In practice, we are usually interested in subsets of Hilb, that exclude degenerate subschemes with very high Castelnuovo-Mumford regularity:

Proposition 3.10. Let Hilb^{∘,m} ⊂ Hilb denote the open subset parametrizing [X] ∈ Hilb satisfying:

• O_{X}(m) has no higher cohomology;

• Sym^{m}V^{∗} ) Γ(O_{X}(m)) is surjective. Let π : X ) Hilb^{∘,m} denote the universal family restricted to this subset. Then we have

det(π_{∗}O_{X} (m)) = Λ_{m}|Hilb^{∘,m} = ∑_{i=0}^{r+1} \binom{m}{i}L_{i}|Hilb^{∘,m}

and there exists a morphism

ϕ_{m} : Hilb^{∘,m} ) Gr(P(m), Sym^{m}V) ⊂ ℙ(⋀^{P(m)} Sym^{m}V)
[X] ) [X]_{m}

such that ϕ_{m}^{∗}O(1) = Λ_{m}.

Remark 3.11. Hilb^{∘,m} contains the open subset parametrizing subschemes X with m-regular ideal sheaf I_{X} and (m − 1)-regular structure sheaf O_{X}. In particular, Hilb^{∘,m} = Hilb for m ≫ 0.

20 BRENDAN HASSETT AND DONGHOON HYEON

Proof. The first assertion is just an application of Equation 3.2 in the special situation when ℝ^{i}π_{∗}O_{X} (m) = 0 for i > 0. The morphism ϕ_{m} is just the classifying map for the surjection of locally free sheaves

Sym^{m}V^{∗} ⊗ O_{Hilb^{∘,m}} ↠ π_{∗}O_{X} (m).

Again, if Q ) Gr(P(m), Sym^{m}V) is the universal quotient bundle then ϕ_{m}^{∗}Q = π_{∗}O_{X} (m), and taking determinants gives the equality of line bundles. □

Applying the functorial properties of the Hilbert-Mumford index [MFK94, 2.1] we obtain:

Corollary 3.12. Retain the notation of Proposition 3.10. Suppose that [X] ∈ Hilb^{∘,m} and ρ : 𝔾_{m} ) GL(V) is a one-parameter subgroup as before such that

lim ; t)0ρ(t).[X] ∈ Hilb^{∘,m}.

Then we have

(3.5)  μ([X]_{m}, ρ) = μ^{Λ_{m}} ([X], ρ).

3.6. Chow stability and Hilbert stability. We compare the geometric invariant theory of the Hilbert points [X]_{m}, m ≫ 0 with that of the Chow point Ch(X).

Let Chow ⊂ ℙ(⊗^{r+1}Sym^{d}V) denote the corresponding Chow variety, i.e., the image of the Hilbert scheme under the morphism [MFK94, §5.4]

\begin{matrix}ϖ : Hilb & ) & Chow ⊂ ℙ(⊗^{r+1}Sym^{d}V) \\ [X] & ) & Ch(X) \end{matrix} .

This is equivariant under the natural actions of SL(V). By [KM76, Theorem 4], we obtain the proportionality

(3.6)  ϖ^{∗}O_{Chow} (1) ∼ L_{r+1}

and thus

(3.7)  lim ; m) \frac{Λ_{m}}{\binom{m}{r+1}} ∼ ϖ^{∗}O_{Chow} (1).

In other words, the sequence {[Λ_{m}]} converges to the pull back of the Chow polarization in the projectivized Néron-Severi group of the Hilbert scheme.

Let

Chow^{s} ⊂ Chow^{ss} ⊂ Chow

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denote the locus of points stable and semistable under the SL(V)-action. For each m ≫ 0, let

Hilb^{s,m} ⊂ Hilb^{ss,m} ⊂ Hilb

the locus of points stable and semistable under the SL(V)-action linearized by Λ_{m}. The ample cone of Hilb admits a finite decomposition into locally-closed cells, such that the semistable locus is constant for linearizations taken from a given cell [DH98, Theorem 0.2.3(i)]. In particular, Hilb^{s,m} and Hilb^{ss,m} are constant for m ≫ 0; these are loci of the points stable and semistable with respect to the asymptotic linearization. While the linearization is not well-defined, its locus of semistable points is!

Moreover, applying functoriality of stability [Rei89, Theorem 2.1], we find

Proposition 3.13. Let X ⊂ ℙ^{N} be a variety. If X is Chow stable then X is m-Hilbert stable for m ≫ 0. If X is m-Hilbert semistable for m ≫ 0 then X is Chow semistable.

Corollary 3.14. Assume the cycle map induces an SL(V)-equivariant map

ϖ : Hilb^{ss,m} ) Chow^{ss},

which is the case for m ≫ 0. Then we obtain a natural morphism of GIT quotients

ϖ : Hilb^{ss,m} //SL(V) ) Chow^{ss}//SL(V).

3.7. Filtered Hilbert polynomials.

Definition 3.15. Given a graded ideal I ⊂ SymV^{∗}, the filtered Hilbert function is defined

H_{SymV^{∗}/I,ρ}(m) = ∑ wt_{ρ}(x^{a})

where the sum is taken over the monomials of degree m not contained in in_{≺_{ρ}}I. For a closed subscheme X ⊂ ℙ(V), we define

H_{X,ρ} = H_{SymV^{∗}/I_{X},ρ}.

Proposition 3.16. The filtered Hilbert function H_{X,ρ}(m) is a polynomial P_{X,ρ}(m) for m ≥ M, the Castelnuovo-Mumford regularity of O_{X}. This polynomial is called the filtered Hilbert polynomial.

Proof. Since m ≥ M we have an embedding

Hilb ⊂ Gr(P(m), Sym^{m}V) ) ℙ( ⋀^{P(m)} Sym^{m}V)

and Proposition 3.10 implies

μ([X]_{m}, ρ) = μ^{Λ_{m}} ([X], ρ).

Equation 3.3 gives

Λ_{m} = ∑_{i=0}^{r+1} \binom{m}{i}L_{i}

Fixing the point and the one-parameter subgroup, μ is a homomorphism in the line bundle variable [MFK94, 2.2] and we have

(3.8)  μ^{Λ_{m}} ([X], ρ) = ∑_{i=0}^{r} \binom{m}{i}μ^{L_{i}}([X], ρ).

The result follows from Equation 3.1. □

3.8. Hilbert schemes of curves. In this section, we assume Hilb parametrizes schemes of pure dimension one. Here Equation 3.2 takes the form

Λ_{m} = L_{0} + mL_{1} + \binom{m}{2}L_{2}.

Proposition 3.17. Let Hilb^{•} ⊂ Hilb denote the open subset parametrizing [X] ∈ Hilb satisfying:

• X is connected of pure dimension one;

• V^{∗} ) Γ(O_{X}(1)) is an isomorphism;

• O_{X} is 2-regular. Then for each m ≥ 2 we have (3.9) μ([X]_{m}, ρ) = μ^{Λ_{m}} ([X], ρ) = (m−1) [(3 − m)μ^{Λ_{2}} ([X], ρ) + (m/2 − 1)μ^{Λ_{3}} ([X], ρ)] .

Proof. Proposition 3.9 gives the relation L_{0}+L_{1} = 0 over Hilb^{•}, and Equation 3.4 gives

Λ_{m}|Hilb^{•} = (m − 1)(L_{1} + \frac{m}{2}L_{2}).

Under our regularity hypothesis, Proposition 3.10 applies for each m ≥ 2 and

(3.10)  μ^{Λ_{m}} ([X], ρ) = (m − 1) [μ^{L_{1}}([X], ρ) + \frac{m}{2}μ^{L_{2}}([X], ρ)]

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is a polynomial for m ≥ 2 (see Proposition 3.16). One can obtain (3.9) by expressing μ^{L_{1}}([X], ρ) and μ^{L_{2}}([X], ρ) in terms of μ^{Λ_{2}} ([X], ρ) and μ^{Λ_{3}} ([X], ρ) and plugging them in (3.10). □

4. Basin of attraction and equivalences

Definition 4.1. Let X be a variety with 𝔾_{m} acting via ρ : 𝔾_{m} ) Aut(X) with fixed points X^{ρ}. For each x^{⋆} ∈ X^{ρ}, the basin of attraction is defined

A_{ρ}(x^{⋆}) := ⟨x ∈ X | lim ; t)0ρ(t).x = x^{⋆}⟩ .

When X is smooth and projective this can be interpreted via the Białynicki-Birula decomposition [BB73, Theorem 4.3]: Consider the decomposition X_{i}^{ρ}, i ∈ I of the fixed points into connected components. Then there is a unique locally closed ρ-invariant decomposition X = ∪_{i∈I}X_{i} and morphisms γ_{i} : X_{i} ) X_{i}^{ρ} such that

• (X_{i})^{ρ} = X_{i}^{ρ} for each i ∈ I;

• γ_{i} is an affine bundle;

• for each x ∈ X_{i}^{ρ}, the tangent space T_{x}X_{i} ⊂ T_{x}X is the subspace over which

ρ acts with nonnegative weights. For x^{⋆} ∈ X_{i}^{ρ} we have A_{ρ}(x^{⋆}) = γ_{i}^{−1}(x^{⋆}).

The importance of this decomposition for the analysis of semistable points is clear from the following proposition which is well known to experts. Given a point on a projective variety x ∈ X ⊂ ℙ^{N}, let x^{∗} ∈ 𝔸^{N+1} denote an affine lift, i.e., a point in the affine cone over X lying over x.

Proposition 4.2. Suppose that G is a reductive linear algebraic group acting on a projective variety X and L is a G-linearized ample line bundle. Suppose x_{1}, x_{2} ∈ X be semistable points mapping to the same point in the GIT quotient X//G. Then there exists a semistable point x_{0} ∈ X with the following properties:

• the orbit Gx_{0}^{∗} is closed, or equivalently, the stabilizer G_{x_{0}^{∗}} ⊂ G is reductive;

• there exists g ∈ G, one-parameter subgroups ρ_{1}, ρ_{2} of G_{x_{0}^{∗}}, and lifts x_{1}^{∗} and x_{2}^{∗} of x_{1} and x_{2} such that

x_{1}^{∗} ∈ A_{ρ_{1}} (x_{0}^{∗}) g · x_{2}^{∗} ∈ A_{ρ_{2}} (x_{0}^{∗}).

Proof. Since x_{1} and x_{2} are identified in the GIT quotient, any homogeneous invariant vanishing on x_{1} automatically vanishes on x_{2}, and vice versa. Consider

24 BRENDAN HASSETT AND DONGHOON HYEON

the orbit closures \overline{Gx_{1}} and \overline{Gx_{2}} in X. Their orbit closures meet [Ses77, Proposition 7, pp. 254]:

\overline{Gx_{1}} ∩ \overline{Gx_{2}} ≠ ∅,

and moreover there exist x_{1}^{∗} and x_{2}^{∗} lying over x_{1} and x_{2} in the affine cone over X such that

\overline{Gx_{1}^{∗}} ∩ \overline{Gx_{2}^{∗}} ≠ ∅.

(This is essentially the fact that invariants separate orbit closures in affine space, e.g., [MFK94, Corollary 1.2, pp. 29].) Pick y_{0}^{∗} ∈ \overline{Gx_{1}^{∗}} ∩ \overline{Gx_{2}^{∗}} generating a closed orbit of the intersection.

Recall Matsushima’s Criterion [Mat60, BB63]: Suppose G is a reductive algebraic group and H ⊂ G a closed subgroup; the homogeneous space G/H is affine if and only if H is reductive. This gives the equivalence of the two conditions on x_{0}^{∗}.

We apply [Kem78, Theorem 1.4] to the closed G-invariant set S = \overline{Gy_{0}^{∗}} = Gy_{0}^{∗}: There exist one-parameter subgroups ρ_{1} and ρ′_{2} such that

x_{0}^{∗} := lim ; t)0ρ_{1}(t) · x_{1}^{∗} ∈ Gy_{0}^{∗} lim ; t)0ρ′_{2}(t) · x_{2}^{∗} ∈ Gy_{0}^{∗}.

Clearly there exists g ∈ G such that

g · lim ; t)0ρ′_{2}(t) · x_{2}^{∗} = x_{0}^{∗}.

Setting ρ_{2} = gρ′_{2}g^{−1}, we obtain the desired result. □

Also, as far as stability is concerned, the points in a basin of attraction are all equivalent if the attracting point is strictly semistable with respect to the 1-PS:

Lemma 4.3. Let G, X, L be as in Proposition 4.2. Let x ∈ X and suppose there exists an x_{0} ∈ X and a one-parameter subgroup ρ of G such that x ∈ A_{ρ}(x_{0}). If μ^{L}(x, ρ) = 0 then x_{0} is semistable with respect to L if and only if x is semistable with respect to L.

Proof. Assume that X is embedded in ℙ^{N} by sections of L and x^{∗} ∈ 𝔸^{N+1} \ {0} be an affine lift of x. Since μ^{L}(x, ρ) = 0, ρ(t).x^{∗} has a specialization, say x_{0}^{∗} ≠ 0, which corresponds to x_{0} ∈ ℙ^{N}. Let s be a G-invariant section of L. Then s(x_{0}^{∗}) = s(ρ(t).x^{∗}) = s(x^{∗}) and it follows that s(x_{0}) ≠ 0 if and only if s(x) ≠ 0. □

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 25

Let x_{0} be a point in X and ρ : 𝔾_{m} ) G be a one-parameter subgroup fixing x_{0}. It follows directly from the definition of the Hilbert-Mumford index that if μ^{L}(x_{0}, ρ) < 0 then every point in the basin of attraction A_{ρ}(x_{0}) is unstable. This observation can be used to classify unstable points in certain situations; our approach is similar to [Tha96, §4]:

Proposition 4.4. Let G, X, L be as in Proposition 4.2 and M a second Glinearized ample line bundle. Let x_{1} ∈ X be semistable with respect to L but unstable with respect to L⊗M^{ϵ} for each rational ϵ > 0. Then there exists a point x_{0} ∈ X having the following properties:

(1) x_{0} is strictly semistable with respect to L;

(2) there exists a one-parameter subgroup ρ : 𝔾_{m} ) G_{x_{0}} such that

x_{1} ∈ A_{ρ}(x_{0});

(3) μ^{L⊗M^{ϵ}} (x_{0}, ρ) < 0.

That is, every strictly semistable point that becomes unstable after perturbing L can be destabilized by a one-parameter subgroup acting via automorphisms of a point strictly semistable with respect to L.

Proof. Let ρ be a one-parameter subgroup with μ^{L⊗M^{ϵ}} (x_{1}, ρ) < 0, which exists by the Hilbert-Mumford criterion. Let x_{0} = lim_{t)0} ρ(t)·x_{1} denote the corresponding limit point in X. Clearly, x_{1} ∈ A_{ρ}(x_{0}) and μ^{L⊗M^{ϵ}} (x_{0}, ρ) < 0. □

For the convenience of the reader, we recall the standard Semistable Replacement Theorem:

Theorem 4.5. Retain the assumptions of Lemma 4.3. Assume that G is reductive so the GIT quotient scheme X^{ss}//G exists. Let B be a smooth curve, 0 ∈ B a closed point, and f : B \ {0} ) X^{ss} be a regular morphism. Then there exists a covering α : B′ ) B branched only over 0 and γ : B′ \ {0′} ) G, 0′ = α^{−1}(0), such that

• there is a regular morphism f′ : B′ ) X^{ss};

• f(α(b′)) = γ(b′).f′(b′) for all b′ ≠ 0′.

Definition 4.6. Two c-semistable curves C_{1} and C_{2} are said to be c-equivalent, denoted C_{1} ∼_{c} C_{2}, if there exists a curve C^{⋆} (which we may assume has reductive automorphism group) and one-parameter subgroups ρ_{1}, ρ_{2} of Aut(C^{⋆})

26 BRENDAN HASSETT AND DONGHOON HYEON

with μ(Ch(C^{⋆}), ρ_{i}) = 0 such that the basins of attraction A_{ρ_{1}} (Ch(C^{⋆})) and A_{ρ_{2}} (Ch(C^{⋆})) contain Chow-points of curves isomorphic to C_{1} and C_{2} respectively.

We define h-equivalence, denoted ∼_{h}, in an analogous way. Lemma 4.3 shows that these equivalence relations respect the semistable and unstable loci. Proposition 4.2 shows that for GIT-semistable curves C_{1} ∼_{c} C_{2} if and only if Ch(C_{1}) and Ch(C_{2}) yield the same point of \overline{M}_{g}^{cs}; the analogous statement holds for h-equivalence.

5. Computations over the moduli space of stable curves

Let π : \overline{C_{g}} ) \overline{M}_{g} denote the universal curve over the moduli stack of curves of genus g. For each n ≥ 1 we have the vector bundle

E_{n} = π_{∗}ω_{π}^{n},

of rank

r(n) = \{ \begin{matrix}g & if n = 1, \\ (2n − 1)(g − 1) & if n > 1. \end{matrix}

Write

λ_{n} = c_{1}(E_{n})

and use λ to designate λ_{1}.

Consider the multiplication maps

(5.1)  Sym^{m}E_{n} ) E_{mn}

for each m > 1. We have the Chern-class identities

c_{1}(Hom(Sym^{m}(E_{n}), E_{mn} )) = rk(Sym^{m}(E_{n}))c_{1}(E_{mn} ) − rk(E_{mn} )c_{1}(Sym^{m}(E_{n}))
= \binom{m + r(n) − 1}{m}λ_{mn} − r(mn)\binom{m + r(n) − 1}{m − 1}λ_{n}
= \frac{(m + r(n) − 1)!}{m! r(n)!} (r(n)λ_{mn} − r(mn)mλ_{n})
∼ r(n)λ_{mn} − r(mn)mλ_{n},

where ∼ designates proportionality. These divisor classes were introduced by Viehweg [Vie89, §1.4] and Cornalba-Harris [CH88, §2] and their significance is explained by the following fact:

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 27

Proposition 5.1. Assume that n ≥ 2 when g = 2. Consider the Hilbert scheme Hilb of degree 2n(g−1) curves of genus g in ℙ^{r(n)−1} . Let Hilb_{g,n} ⊂ Hilb denote the closure of the n-pluricanonically embedded smooth curves of genus g. Suppose that Λ_{m} (introduced in §3.2) is well-defined and ample on Hilb_{g,n}.

Consider the open subsets

V_{g,n}^{s,m} ⊂ Hilb_{g,n}^{s,m} ⊂ Hilb_{g,n} , V_{g,n}^{ss,m} ⊂ Hilb_{g,n}^{ss,m} ⊂ Hilb_{g,n}

corresponding to n-canonically embedded Deligne-Mumford stable curves that are GIT stable and GIT semistable with respect to Λ_{m}. Let

U_{g,n}^{s,m} ⊂ U_{g,n}^{ss,m} ⊂ \overline{M}_{g}

denote their images in moduli. Then Λ_{m} descends to a multiple of r(n)λ_{mn} − r(mn)mλ_{n} along U_{g,n}^{ss,m} . This restricts to an ample divisor on the coarse moduli space U_{g,n}^{s,m}.

Proof. (cf. [Vie95, §1.6]) We illustrate how Λ_{m} descends to U_{g,n}^{ss,m} . Let ϖ : X ) Hilb_{g,n} denote the universal family. The multiplication map (5.1) on moduli is obtained by descent from the multiplication map over Hilb_{g,n}

Sym^{m}(ϖ_{∗}O_{X} (1)) ) ϖ_{∗}O_{X} (m).

As in Proposition 3.9, we have a trivialization

Γ(O_{ℙ^{r(n)−1}} (1)) ⊗ O_{V_{g,n}^{ss,m}} ≃ ϖ_{∗}O_{X} (1)|V_{g,n}^{ss,m} .

Thus the divisor class

c_{1} (Hom(Sym^{m}(ϖ_{∗}O_{X} (1)), ϖ_{∗}O_{X} (m))|V_{g,n}^{ss,m} )

is proportional to

c_{1}(ϖ_{∗}O_{X} (m)|V_{g,n}^{ss,m} ) = Λ_{m}|V_{g,n}^{ss,m} .

As for the ampleness, the coarse moduli space U_{g,n}^{s,m} of U_{g,n}^{s,m} can be identified with an open subset of the GIT quotient

Hilb_{g,n}^{ss,m} //SL_{r(n)} .

Λ_{m} descends to a polarization of this quotient. □

Mumford [Mum77, Theorem 5.10] showed that the Grothendieck-Riemann-Roch formula gives

(5.2)  λ_{n} = (6n^{2} − 6n + 1)λ − \binom{n}{2}δ, n > 1.

We therefore find

(5.3)

r(n)λ_{mn} − r(mn)mλ_{n} = \{ \begin{matrix}λ + (m − 1)((4g + 2)m − g + 1)λ − \frac{gm}{2} δ) if n = 1, \\ (m − 1)(g − 1)((6mn^{2} − 2mn − 2n + 1)λ − \frac{mn^{2}}{2} δ) if n > 1. \end{matrix}

Asymptotically as m ) , we obtain the proportionality

lim ; m) (r(n)λ_{mn} − r(mn)mλ_{n}) ∼ \{ \begin{matrix}(4g + 2)λ − \frac{g}{2}δ if n = 1 \\ (6n − 2)λ − \frac{n}{2}δ if n > 1. \end{matrix}

Combining Proposition 5.1 and Equation 3.7, we obtain:

Proposition 5.2. Assume that n ≥ 2 when g = 2. Consider the Chow variety Chow of degree 2n(g − 1) curves of genus g in ℙ^{r(n)−1} . Let Chow_{g,n} ⊂ Chow denote the closure of the n-pluricanonically embedded curves of genus g.

Consider the open subsets

V_{g,n}^{s,} ⊂ Chow_{g,n}^{s} ⊂ Chow_{g,n} , V_{g,n}^{ss,} ⊂ Chow_{g,n}^{ss} ⊂ Chow_{g,n}

corresponding to n-canonically embedded Deligne-Mumford stable curves that are Chow stable and Chow semistable respectively. Let

U_{g,n}^{s,} ⊂ U_{g,n}^{ss,} ⊂ \overline{M}_{g}

denote their images in moduli. Then the polarization descends to a multiple of

(4g + 2)λ − \frac{g}{2}δ if n = 1

or

(6n − 2)λ − \frac{n}{2}δ if n > 1.

The restriction to the coarse moduli space U_{g,n}^{s,} is ample.

Remark 5.3 (Application to polarizations on \overline{M}_{g}). Mumford has proven that U_{g,n}^{s,} = \overline{M}_{g} for each n ≥ 5 [Mum77, Theorem 5.1]. Proposition 3.13 then guarantees that U_{g,n}^{s,m} = \overline{M}_{g} for all m ≫ 0. Proposition 5.2 then implies that aλ − δ is ample for a > 11.2 [Mum77, Corollary 5.18]. Cornalba and Harris [CH88] established the sharp result: aλ − δ is ample if and only if a > 11.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 29

We are primarily interested in situations where not all Deligne-Mumford stable curves have stable Hilbert/Chow points. Here GIT yields alternate birational models of the moduli space.

Theorem 5.4. Retain the notation of Propositions 5.1 and 5.2 with the convention that m = in the Chow case. Suppose that

• the complement to the Deligne-Mumford stable curves in the GIT-semistable locus Hilb_{g,n}^{ss,m} (resp. Chow_{g,n}^{ss}) has codimension ≥ 2;

• there exist Deligne-Mumford stable curves in the GIT-stable locus Hilb_{g,n}^{s,m}

(resp Chow_{g,n}^{s}). Then there exists a birational contraction

F : \overline{M}_{g} −−→ Hilb_{g,n}^{ss,m} //SL_{r(n)} (resp. Chow_{g,n}^{ss} //SL_{r(n)} )

regular along the Deligne-Mumford stable curves with GIT-semistable Hilbert (resp. Chow) points.

If L_{m} is the polarization on the GIT quotient induced by Λ_{m} then the moving divisor

(5.4)  F^{∗}L_{m} ∼ r(n)λ_{mn} − r(mn)mλ_{n} (mod Exc(F)),

where Exc(F) ⊂ Pic(\overline{M}_{g}) is the subgroup generated by F-exceptional divisors.

A rational map of proper normal varieties is said to be a birational contraction if it is birational and its inverse has no exceptional divisors. Note that Propositions 5.1 and 5.2 cover the case where F is an isomorphism.

Proof. Our assumptions can be written in the notation of Propositions 5.1 and 5.2:

• V_{g,n}^{ss,m} ⊂ Hilb_{g,n}^{ss,m} (resp. V_{g,n}^{ss,} ⊂ Chow_{g,n}^{ss}) has codimension ≥ 2;

• V_{g,n}^{s,m} ≠ ∅. The GIT quotient morphism

V_{g,n}^{s,m} ) U_{g,n}^{s,m}

identifies the stack-theoretic quotient [V_{g,n}^{s,m} /SL_{r(n)} ] with U_{g,n}^{s,m}. This gives a birational map

Hilb_{g,n}^{ss,m} //SL_{r(n)} (resp. Chow_{g,n}^{ss} //SL_{r(n)} ) −−→ \overline{M}_{g};

we define F as its inverse.

30 BRENDAN HASSETT AND DONGHOON HYEON

We establish that F is regular along U_{g,n}^{ss,m} : We have an SL_{r(n)} -equivariant morphism

Hilb_{g,n}^{ss,m} ) Hilb_{g,n}^{ss,m} //SL_{r(n)} ,

which descends to

U_{g,n}^{ss,m} ) Hilb_{g,n}^{ss,m} //SL_{r(n)} .

Recall the universal property of the coarse moduli space: Any morphism from a stack to a scheme factors through its coarse moduli space. In our context, this gives

U_{g,n}^{ss,m} ) Hilb_{g,n}^{ss,m} //SL_{r(n)} (resp. Chow_{g,n}^{ss} //SL_{r(n)} ).

Furthermore, the total transform of \overline{M}_{g} \ U_{g,n}^{ss,m} is contained in the complement Hilb_{g,n}^{ss,m} \ V_{g,n}^{ss,m} (resp. Chow_{g,n}^{ss} \ V_{g,n}^{ss,} ), which has codimension ≥ 2. Thus any divisorial components of \overline{M}_{g} \ U_{g,n}^{ss,m} are F-exceptional divisors. Similary, F^{−1} has no exceptional divisors: These would give rise to divisors in the complement to V_{g,n}^{ss,m} in the semistable locus.

We now analyze F^{∗}L_{m} in the rational Picard group of \overline{M}_{g}. (Since \overline{M}_{g} has quotient singularities, its Weil divisors are all ℚ-Cartier.) If L_{m}^{a} is very ample on the GIT quotient then F^{∗}L_{m}^{a} induces F, i.e., F^{∗}L_{m}^{a} has no fixed components and is generated by global sections over U_{g,n}^{ss,m} . Now F^{∗}L_{m} is proportional to r(n)λ_{mn} − r(mn)mλ_{n} over U_{g,n}^{ss,m} and Formula (5.4) follows. □

6. Properties of c-semistable and h-semistable curves

6.1. Embedding c-semistable curves.

Proposition 6.1. If g ≥ 3 and C is a c-semistable curve of genus g over k, then H^{1}(C, ω_{C}^{⊗n}) = 0 and ω_{C}^{⊗n} is very ample for n ≥ 2.

Remark 6.2. For the rest of this paper, when we refer to the Chow or Hilbert point of a c-semistable curve C it is with respect to its bicanonical embedding in ℙ(Γ(C, ω_{C}^{⊗2})^{∗}).

Proof. Our argument follows [DM69, Theorem 1.2].

By Serre Duality, H^{1}(C, ω_{C}^{⊗n}) vanishes if H^{0}(C, ω_{C}^{⊗1−n}) vanishes. The restriction of ω_{C}^{⊗1−n} to each irreducible component D ⊂ C has negative degree because ω_{C} is ample. It follows that Γ(D, ω_{C}^{⊗1−n}|D) = 0, hence Γ(C, ω_{C}^{⊗1−n}) = 0.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 31

To show that ω_{C}^{⊗n} is very ample for n ≥ 2, it suffices to prove for all x, y ∈ C that

(6.1)  Hom(𝔪_{x}𝔪_{y}, ω_{C}^{⊗−n}) = 0, n ≥ 1.

Let π : C′ ) C denote the partial normalization of any singularities at x and y. When x is singular, a local computation gives

Hom(𝔪_{x}, L) ≃ Γ(C′, π^{∗}L).

If x is a cusp and x′ ∈ C′ its preimage then

Hom(𝔪_{x}^{2}, L) ≃ Γ(C′, π^{∗}L(2x′)).

If x is a node or tacnode and x_{1}, x_{2} ∈ C′ the preimage points then

Hom(𝔪_{x}^{2}, L) ≃ Γ(C′, π^{∗}L(x_{1} + x_{2})).

Thus in each case we can express

Hom(𝔪_{x}𝔪_{y}, ω_{C}^{⊗−n}) = Γ(C′, M)

for a suitable invertible sheaf M on C′. Moreover, we have an inclusion

π^{∗}ω_{C}^{−n} ) M

with cokernel Q supported in π^{−1}{x, y} of length ℓ(Q) ≤ 2. For instance, if both x and y are smooth then

M = ω_{C}^{−n}(x + y);

if both x and y are singular and x ≠ y then

M = π^{∗}ω_{C}^{−n}.

Suppose that for each irreducible component D′ ⊂ C′, the degree deg M|D′ < 0. Then Γ(C′, M) = 0 and the desired vanishing follows. We therefore classify situations where

deg M|D′ = −ndeg π^{∗}ω_{C}|D′ + ℓ(Q|D′) ≥ 0,

which divide into the following cases:

(a) deg π^{∗}ω_{C}|D′ = 1, n = 1, ℓ(Q|D′) = 1;

(b) deg π^{∗}ω_{C}|D′ = 1, n = 1, 2, ℓ(Q|D′) = 2;

(c) deg π^{∗}ω_{C}|D′ = 2, n = 1, ℓ(Q|D′) = 2.

32 BRENDAN HASSETT AND DONGHOON HYEON

We write D = π(D′) ⊂ C.

We enumerate the various possibilities. We use the assumption that C is csemistable and thus has no elliptic tails. In cases (a) and (b), D is necessarily isomorphic to ℙ^{1} and meets the rest of C in either three nodes or in one node and one tacnode. After reordering x and y, we have the following subcases:

(a1) x = y ∈ D a node or tacnode of C;

(a2) x ∈ D a node or tacnode of X and y ∈ D a smooth point of C; (a3) x ∈ D a smooth point of C and y ∉ D. (b1) x, y ∈ D smooth points of C.

In case (c), D may have arithmetic genus zero or one:

(c1) D ≃ ℙ^{1} with x, y ∈ D smooth points of C;

(c2) D of arithmetic genus one with x, y ∈ D smooth points of C; (c3) D of arithmetic genus one, x = y a node or cusp of D, and D′ ≃ ℙ^{1}.

In subcase (c1), D meets the rest of C in either four nodes, or in two nodes and one tacnode, or in two tacnodes. In subcases (c2) and (c3), D meets the rest of C in two nodes. Except in case (c3), π : D′ ) D is an isomorphism.

For subcases (b1), (c1), and (c2), π is an isomorphism. Moreover, Q is supported along D so M has negative degree along any other irreducible components of C. There are other components because the genus of C is at least three. Thus elements of Γ(C, M) restrict to elements of Γ(D, M|D) that vanish at the points where D meets the other components, i.e., in at least two points. Since deg M|D = 0 or 1, we conclude Γ(C, M) = 0.

For subcase (c3), π is not an isomorphism but Q is still supported along D′. As before, M has negative degree along other irreducible components of C′, and elements of Γ(C′, M) restrict to elements of Γ(D′, M|D′) vanishing where D′ meets the other components. There are at least two such points but deg M|D′ = 0, 1, so we conclude that Γ(C′, M) = 0.

In case (a), we have deg M|D = 0. Subcases (a1) and (a2) are similar to (b1) and (c1): Q is supported along D′ so elements in Γ(C′, M) restrict to elements of Γ(D′, M|D′) vanishing at the points where D′ meets the other components. There is at least one such point, e.g., the singularity not lying over x, hence Γ(C′, M) = 0.

Subcase (a3) is more delicate. If D′ is the unique component such that deg (M|D′) ≥ 0 then the arguments of the previous cases still apply. However,

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 33

the support of Q might not be confined to a single component. We suppose there are two components D′_{1} and D′_{2} as described in (a3), such that deg (M|D′_{i}) ≥ 0. Since the genus of C is > 2, C cannot just be the union of D′_{1} and D′_{2}; there is at least one additional component meeting each D′_{i} at some point z_{i}, and the restriction of M to this component has negative degree. Thus elements of Γ(C′, M) restrict to elements of Γ(D′_{i}, M|D′_{i}) vanishing at z_{i}, which are necessarily zero. □

Corollary 6.3. Let C ⊂ ℙ^{3g−4} be a c-semistable bicanonical curve.

• O_{C} is 2-regular.

• The Hilbert scheme is smooth at [C].

• Let p_{1}, …, p_{n} denote the singularities of C and Def(C, p_{i}), i = 1, …, n their versal deformation spaces. Then there exists a neighborhood U of [C] in the Hilbert scheme such that

U ) ∏_{i} Def(C, p_{i})

is smooth.

Proof. Proposition 6.1 yields

H^{1}(C, O_{C}(1)) = H^{1}(C, ω_{C}^{⊗2}) = 0

which gives the regularity assertion. This vanishing also implies [Kol96, I.6.10.1]

H^{1}(C, Hom(I_{C}/I_{C}^{2}, O_{C})) = 0;

since the singularities of C are local complete intersections we have

Ext^{1}(I_{C}/I_{C}^{2}, O_{C}) = H^{1}(C, Hom(I_{C}/I_{C}^{2}, O_{C})) = 0

thus the Hilbert scheme is unobstructed at [C] (see [Kol96, I.2.14.2]). The assertion about the map onto the versal deformation spaces is [Kol96, I.6.10.4]. □

Corollary 6.4. Let C ⊂ ℙ^{3g−4} be a bicanonical c-semistable curve and C^{⋆} denote the curve to which ρ(t).C specializes. If C^{⋆} is a bicanonical c-semistable curve then

μ([C]_{m}, ρ) = (m − 1) [(3 − m)μ([C]_{2}, ρ) + (m/2 − 1)μ([C]_{3}, ρ)] .

Thus [C]_{m} is stable (resp. strictly semistable, resp. unstable) with respect to ρ for each m ≥ 2 if and only if μ([C]_{3}, ρ) ≥ 2μ([C]_{2}, ρ) > 0 (resp. μ([C]_{3}, ρ) =

34 BRENDAN HASSETT AND DONGHOON HYEON

μ([C]_{2}, ρ) = 0, resp. μ([C]_{3}, ρ) ≤ 2μ([C]_{2}, ρ) < 0.) The Chow point Ch(C) is stable (resp. strictly semistable, resp. unstable) with respect to ρ if and only if μ([C]_{3}, ρ) − 2μ([C]_{2}, ρ) > 0 (resp. = 0, resp. < 0.)

Proof. By Proposition 6.1, a bicanonical c-semistable is 2-regular and the assertion on the Hilbert points follows immediately from Equation (3.9). Equations (3.6) and (3.8) allow us to interpret the Hilbert-Mumford index of the Chow point in terms of the leading coefficient of μ([C]_{m}, ρ) as a polynomial in m. □

6.2. Basic properties of tacnodal curves. Let C be a curve with a tacnode r, i.e., a singularity with two smooth branches intersecting with simple tangency. Let ν : D ) C be the partial normalization of C at r and ν^{−1}(r) = {p, q} ⊂ D the conductor. The descent data from (D, p, q) to (C, r) consists of a choice of isomorphism

ι : T_{p}D )^{∼} T_{q}D

identifying the tangent spaces to the branches. Functions on C pull back to functions f on D satisfying f(p) = f(q) and ι(f′(p)) = f′(q).

Varying the descent data gives a one-parameter family of tacnodal curves:

Proposition 6.5. Let D be a reduced curve and p, q ∈ D distinct smooth points with local parameters σ_{p} and σ_{q}. Each invertible linear transformation T_{p}D ) T_{q}D can be expressed

ι(t) : \frac{∂}{∂σ_{p}} ) t\frac{∂}{∂σ_{q}}

for some t ≠ 0; let 𝔾_{m} ≃ Isom(T_{p}D, T_{q}D) denote the corresponding identification. Then there exists a family C ) 𝔾_{m}, a section r : 𝔾_{m} ) C, and a morphism

\xymatrix{D × 𝔾_{m} \ar[rr]^{ν} \ar[dr] &  & C \ar[dl] \\  & 𝔾_{m}}

such that (1) ν restricts to an isomorphism

D \ {p, q} × 𝔾_{m} )^{∼} C \ r;

(2) for each t ∈ 𝔾_{m}, r_{t} ∈ C_{t} is a tacnode and ν_{t} its partial normalization;

(3) the descent data from (D, p, q) to (C_{t}, r_{t}) is given by ι(t).

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 35

Every tacnodal curve normalized by (D, p, q) occurs as a fiber of C ) 𝔾_{m}.

If D is projective of genus g − 2 then each C_{t} has genus g.

We sketch the construction of C: ι(t) tautologically yields an identification over 𝔾_{m}

(6.2)  ι : T_{p×𝔾_{m}} D × 𝔾_{m}/𝔾_{m} −)^{∼} T_{q×𝔾_{m}} D × 𝔾_{m}/𝔾_{m}

which is the descent data from D × 𝔾_{m} to C. Fiber-by-fiber, we get the universal family of tacnodal curves normalized by (D, p, q).

We will extend C ) 𝔾_{m} to a family of tacnodal curves C′ ) ℙ^{1}. First, observe that the graph construction gives an open embedding

𝔾_{m} ≃ Isom(T_{p}D, T_{q}D) ⊂ ℙ(T_{p}D ⊕ T_{q}D) ≃ ℙ^{1},

where t = 0 corresponds to [1, 0] and t = corresponds to [0, 1]. However, the identification (6.2) fails to extend over all of ℙ^{1}; indeed, it is not even defined at p × [0, 1] and its inverse is not defined at q × [1, 0]. We therefore blow up

D′ = Bl_{p×[0,1],q×[1,0]} D × ℙ^{1}

and consider the sections

𝔭, 𝔮 : ℙ^{1} ) D′

extending p × 𝔾_{m} and q × 𝔾_{m}. Now (6.2) extends to an identification

ι′ : T_{𝔭}D′/ℙ^{1} −)^{∼} T_{𝔮}D′/ℙ^{1}.

Proposition 6.6. Retain the notation of Proposition 6.5. There exists an extension

C ⊂ C′
]  ]
Isom(T_{p}D, T_{q}D) ≃ 𝔾_{m} ⊂ ℙ^{1} ≃ ℙ(T_{p}D ⊕ T_{q}D)

where C′ ) ℙ^{1} denotes the family of curves obtained from D′ and ι′ by descent, r′ : ℙ^{1} ) C′ the tacnodal section, and ν′ : D′ ) C′ the resulting morphism. The new fiber (C′_{0}, r′(0)) (resp. (C′_{}, r′( ))) is normalized by (D′_{0} = D∪_{q} ℙ^{1}, p, 𝔮(0)) (resp. (D′_{}= D ∪_{p} ℙ^{1}, 𝔭( ), q).)

We say that the tacnodes in the family {C′_{t}, r_{t}}_{t∈ℙ^{1}} are compatible, and that two curves are compatible if one can be obtained from the other by replacing some tacnodes by compatible tacnodes.

36 BRENDAN HASSETT AND DONGHOON HYEON

7. Unstable bicanonical curves

In this section, we show that if a curve is not c-semistable then it has unstable Chow point:

Proposition 7.1. If Ch(C) ∈ Chow_{g,2} is Chow semistable then C ⊂ ℙ^{3g−4} is c-semistable.

We prove this by finding one-parameter subgroups destabilizing curves that are not c-semistable. Many statements in this section are fairly direct generalizations of results in [Mum77] and [Sch91].

7.1. Badly singular curves are Chow unstable. A Chow semistable bicanonical curve C cannot have a triple point, since \frac{d}{N+1} = \frac{4g−4}{3g−3} < \frac{3}{2} and this implies that C is Chow unstable by Proposition 3.1 of [Mum77]. We need to show that among the double points, only nodes, ordinary cusps and tacnodes are allowed.

Lemma 7.2. If C has a non-ordinary cusp, then it is Chow unstable.

Proof. Suppose that C has a non-ordinary cusp at p. Let ν : C̃ ) C be the normalization, p′ = ν^{−1}(p) and assume p = [1, 0, …, 0]. Recall that the singularity at p is determined by the vanishing sequence (a_{i}(ν^{∗}|ω_{C}^{⊗2}|, p′)) _{i=1}^{N+1} which is the strictly increasing sequence determined by the condition

{a_{i}(ν^{∗}|ω_{C}^{⊗2}|, P) | i = 1, 2, …, N + 1} = {ord_{p′}(σ) | σ ≠ 0 ∈ ν^{∗}|ω_{C}^{⊗2}|}.

C has a cusp at p if and only if the vanishing sequence (a_{i}(ν^{∗}|ω_{C}^{⊗2}|, p′)) is of the form (0, 2, ≥ 3), and it has an ordinary cusp if it is of the form (0, 2, 3, ≥ 4).

Hence if C has a non-ordinary cusp at p, then we can choose coordinates x_{0}, …, x_{N} such that ord_{p′}x_{0} = 0, ord_{p′}x_{1} = 2, ord_{p′}x_{2} = 4, and ord_{p′}x_{i} ≥ 5, i = 3, 4, …, N. Let ρ : 𝔾_{m} ) GL_{N+1}(k) be the one-parameter subgroup such that ρ(t).x_{i} = t^{r_{i}} x_{i}, where the weights are:

(r_{0}, r_{1}, …, r_{N}) = (5, 3, 1, 0, …, 0).

Then ord_{p′}x_{i} + r_{i} ≥ 5 for all i, and it follows from Lemma 3.3 that

e_{ρ}(C) = e_{ρ}(C̃) ≥ e_{ρ}(C̃)_{p′} ≥ 5^{2} = 25,

while \frac{2d}{N+1} ∑ r_{i} = \frac{2·4(g−1)}{3(g−1)} · 9 = 24. The assertion now follows from Theorem 3.2. □

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 37

Lemma 7.3. Suppose C has a singularity at p such that

Ô_{C,p} ≃ k[x, y]/(y^{2} − x^{2s}), s ≥ 3.

Then C is unstable.

Proof. Let ν : C̃ ) C be the normalization, ν^{−1}(p) = {p_{1}, p_{2}}. Since the two branches of C agree to order s at p, we may choose coordinates x_{0}, …, x_{N} such that

(ord_{p_{i}} x_{0}, …, ord_{p_{i}} x_{N}) = (0, 1, 2, ≥ 3), i = 1, 2.

Let ρ be the one-parameter subgroup of GL_{N+1}(k) with weights (r_{0}, …, r_{N}) = (3, 2, 1, 0, …, 0). Then we have

ord_{p_{i}} x_{j} + r_{j} ≥ 3, i = 1, 2 and j = 0, 1, …, N,

and by Lemma 3.3,

e_{ρ}(C) = e_{ρ}(C̃) ≥ e_{ρ}(C̃)_{p_{1}} + e_{ρ}(C̃)_{p_{2}} ≥ 2 · 3^{2} = 18,

which is strictly greater than \frac{2d}{N+1} ∑ r_{i} = \frac{2·4(g−1)}{3(g−1)} · 6 = 16.

□

Lemma 7.4. If C has a multiple component, C is Chow unstable.

Proof. Let C_{1} be a component of C with multiplicity n ≥ 2. Choose a smooth non-flex point p ∈ C_{1}^{red} such that p does not lie in any other component. Since p is smooth on C_{1}^{red}, we may choose coordinates x_{0}, …, x_{N} such that

(ord_{p}x_{0}, …, ord_{p}x_{N}) = (0, 1, 2, ≥ 3).

Let ρ be the one-parameter subgroup of GL_{N+1}(k) with weights (r_{0}, …, r_{N}) = (3, 2, 1, …, 0). Then we have

ord_{p}x_{i} + r_{i} ≥ 3.

This yields the inequality

e_{ρ}(C) ≥ n · e_{ρ}(C_{1}) ≥ 2 · 3^{2} = 18

whereas \frac{2d}{N + 1} ∑ r_{i} = \frac{8}{3} · 6 = 16. □

38 BRENDAN HASSETT AND DONGHOON HYEON

7.2. Polarizations on semistable limits of bicanonical curves. We first prove that the semistable limit of a one parameter family of smooth bicanonical curves is bicanonical:

Proposition 7.5. Let C ) Spec k[[t]] be a family of Chow semistable curves of genus g such that the generic fibre C_{η} is smooth. If Φ : C ) ℙ_{k[[t]]}^{3g−4} is an embedding such that Φ_{η}^{∗}(O(1)) = ω_{C_{η} /k[[t]]}^{⊗2} then O_{C}(1) = ω_{C/k[[t]]}^{⊗2}.

By [Mum77, 4.15], nonsingular bicanonical curves are Chow stable. Hence any Chow semistable curve is a limit of nonsingular bicanonical curves and Proposition 7.5 implies that if C is not bicanonical, then Ch(C) ∉ Chow_{g,2}^{ss}. In particular, a Chow semistable curve does not have a smooth rational component meeting the rest of the curve in < 3 points. Mumford proved the statement for the n-canonical curves for n ≥ 5, and his argument can be easily modified to suit our purpose. It is an easy consequence of (ii) of the following proposition, which, in Mumford’s words, says that the degrees of the components of C are roughly in proportion to their natural degrees.

Proposition 7.6 (Proposition 5.5, [Mum77]). Let C ⊂ ℙ^{3g−4} be a connected curve of genus g and degree 4g − 4. Then

(i) C is embedded by a non-special complete linear system.

(ii) Let C = C_{1} ∪ C_{2} be a decomposition of C into two sets of components such that W = C_{1} ∩ C_{2} and w = #W (counted with multiplicity). Then

| deg C_{1} − 2 deg_{C_{1}} ω_{C} | ≤ \frac{w}{2}.

Mumford’s argument goes through in the bicanonical case except for the proof of H^{1}(C_{1}, O_{C_{1}} (1)) = 0. If H^{1}(C_{1}, O_{C_{1}} (1)) ≠ 0, then by Clifford’s theorem we have

h^{0}(C_{1}, O_{C_{1}} (1)) ≤ \frac{deg (C_{1})}{2} + 1

and the Chow semistability of C forces

w + 2 deg C_{1} ≤ \frac{2 deg C}{3g − 3} h^{0}(C_{1}, O_{C_{1}} (1)).

Combining the two, we obtain

deg C_{1} ≤ 4 − \frac{3}{2}w.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 39

If w ≠ 0, then deg C_{1} ≤ 2, hence C_{1} is rational and H^{1}(C_{1}, O_{C_{1}} (1)) = 0. If w = 0, then deg C_{1} ≤ 4 which is absurd since C_{1} = C and deg C_{1} = 4g − 4.

We need to justify our use of Clifford’s theorem here, as Chow semistable bicanonical curves have cusps and tacnodes. We shall sketch the proof of Gieseker and Morrison [Gie82] and highlight the places where modifications are required to accommodate the worse singularities ([Gie82] assumes that C has only nodes).

Theorem 7.7 (Clifford’s Theorem). Let C ⊂ ℙ^{N} be a reduced curve with nodes, cusps and tacnodes. Let L be a line bundle generated by sections. If H^{1}(C, L) ≠ 0, then there is a subcurve C_{1} ⊂ C such that

h^{0}(C, L) ≤ \frac{deg_{C_{1}} L}{2} + 1.

Sketch proof. Suppose that H^{1}(C, L) ≠ 0 and φ ≠ 0 ∈ Hom(L, ω_{C}). Let C_{1} be the union of components where φ does not vanish entirely and p_{1}, …, p_{w} be the intersection points of C_{1} and \overline{C − C_{1}}. Assume that p_{i}’s are ordered so that p_{1}, …, p_{ℓ} are tacnodes. Then we have

ω_{C}|_{C_{1}} (−2 ∑_{i=1}^{ℓ} p_{i} − ∑_{i=ℓ+1}^{w} p_{i}) = ω_{C_{1}} .

We claim that φ restricts to give a homomorphism from L_{C_{1}} to ω_{C_{1}}: Let p_{i} be a tacnode and let D ⊄ C_{1} be the irreducible component containing p_{i}. Since φ vanishes entirely on D, φ must vanish to order ≥ 2 at p_{i} on C_{1}. Likewise, φ must vanish at each node. It follows that φ|_{C_{1}} factors through ω_{C}|_{C_{1}} (−2 ∑ _{i=1}^{ℓ} p_{i} − ∑ _{i=ℓ+1}^{w} p_{i}). Let s_{1}, …, s_{r} be a basis of Hom(L_{C_{1}} , ω_{C_{1}} ) such that s_{1} = φ, and let t_{1}, …, t_{p} be a basis for H^{0}(C, L) such that t_{1} does not vanish at the support of s_{1} and at any singular points. It is shown in [Gie82] that

\begin{aligned}[s_{1}, t_{1}], & [s_{1}, t_{2}], & [s_{1}, t_{3}], & …, & [s_{1}, t_{p}], \\ & [s_{2}, t_{1}], & [s_{3}, t_{1}], & …, & [s_{r}, t_{1}] \end{aligned}

are linearly independent sections of H^{0}(C_{1}, ω_{C_{1}} ), which implies that p + r − 1 ≤ p_{a}(C_{1})+1. Combining it with the Riemann-Roch gives the desired inequality. □

7.3. Elliptic subcurves meeting the rest of the curve in one point. Let C be a Deligne-Mumford stable curve with an elliptic tail E ⊂ C. Then ω_{C}^{⊗2} fails to be very ample along E and thus C does not admit a bicanonical embedding. In particular, C does not arise in GIT quotients of the Chow variety/Hilbert scheme of bicanonical curves.

Here, we focus on curves with an elliptic subcurve meeting the rest of the curve in a tacnode.

Proposition 7.8. Let C = E ∪_{p} R ∪_{q} D be a bicanonical curve consisting of a rational curve E with one cusp, a rational curve R and a genus g − 2 curve D such that p is a tacnode and q is a node. Then C is Chow unstable with respect to a one-parameter subgroup coming from its automorphism group.

Proof. Restricting ω_{C}^{⊗2} we get

ω_{C}^{⊗2}|E ≃ O_{E}(4p), ω_{C}^{⊗2}|R ≃ O_{R}(2), ω_{C}^{⊗2}|D ≃ ω_{D}^{⊗2}(2q).

Since h^{0}(ω_{C}^{⊗2}|D) = 3(g − 2) − 3 + 2 = 3g − 7, we can choose coordinates so that

E ∪_{p} R ⊂ {x_{6} = x_{7} = ⋯ = x_{3g−4} = 0}

and D ⊂ {x_{0} = x_{1} = x_{2} = x_{3} = 0}. E and R can be parametrized by

[s, t] ) [s^{4}, s^{2}t^{2}, st^{3}, t^{4}, 0, …, 0]

and

[u, v] ) [0, 0, uv, u^{2}, v^{2}, 0, …, 0].

The cusp is at [1, 0, …, 0], p = [0, 0, 0, 1, 0, …, 0], and q = [0, 0, 0, 0, 1, 0, …, 0]. Let ρ be the 1-PS with weight (0, 2, 3, 4, 2, …, 2). We have

e_{ρ}(C) ≥ e_{ρ}(E)_{p} + e_{ρ}(R)_{p} + e_{ρ}(R)_{q} + e_{ρ}(D).

On E (and R), we have v_{p}(x_{i}) + r_{i} ≥ 4 for all i where v_{p} is the valuation of O_{E,p} (and O_{R,p} respectively) and r_{i} are weights of ρ. By Lemma 3.3, e_{ρ}(E)_{p} ≥ 4^{2} and e_{ρ}(R)_{p} ≥ 4^{2}. On R, v_{q}(x_{i}) + r_{i} ≥ 2 and e_{ρ}(R)_{q} ≥ 2^{2}. Since ρ acts trivially on D with weight 2, we use Lemma 3.4 and obtain

e_{ρ}(D) = 2 · 2 · deg D = 4(4g − 10).

Combining them all, we obtain

e_{ρ}(C) ≥ 36 + 16g − 40 > 2 · \frac{4}{3} ∑_{i=1}^{3g−4} r_{i} = 16g − \frac{40}{3}.

□

Corollary 7.9. Let C′ = E′ ∪_{p} D be a bicanonical curve consisting of a genus one curve E′ and a genus g − 2 curve D meeting in one tacnode p. Then C′ is Chow unstable.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 41

Proof. In view of Proposition 7.8, it suffices to show that C′ is in the basin of attraction of E ∪_{p} R ∪_{q} D with respect to ρ. Consider the induced action on the local versal deformation space of the cusp [1, 0, …, 0] which is given by

y^{2} = x^{3} + ax + b

where y = x_{2}/x_{0} and x = x_{1}/x_{0}. The 𝔾_{m} action is given by

t.(a, b) = (t^{4}a, t^{6}b)

and the basin of attraction contains arbitrary smoothing of the cusp. On the other hand, the local versal deformation space of the tacnode p = [0, 0, 0, 1, 0, …, 0] is given by

y^{2} = x^{4} + ax^{2} + bx + c

where x = x_{2}/x_{3} so that 𝔾_{m} acts on (a, b, c) with weight (−2, −3, −4) and the basin of attraction does not contain any smoothings of the tacnode. At the node q = [0, 0, 0, 0, 1, 0, …, 0], the local versal deformation space is

xy = c_{0}

where x may be taken to be x_{2}/x_{4} and 𝔾_{m} acts with weight +1 on the branch of R and trivially on D. Thus the induced action on the deformation space has weight +1, and the basin of attraction contains arbitrary smoothing of the node. □

7.4. Hilbert unstable curves. Let C be a bicanonical curve. By Proposition 3.13, C is Chow semistable if it is Hilbert semistable. Note that by definition, if C does not admit an elliptic chain, then C is c-semistable if and only if it is h-semistable. Combining this with Proposition 7.1, we obtain:

Proposition 7.10. If a bicanonical curve is Hilbert semistable and does not admit an elliptic chain, then it is h-semistable.

We shall have completed the implication

Hilbert semistable } h-semistable

once we prove that a Hilbert semistable curve does not admit an elliptic chain. We accomplish this in Proposition 10.4 and Corollary 10.9.

8. Classification of curves with automorphisms

In this section, we classify c-semistable curves with infinite automorphisms.

8.1. Rosaries.

Definition 8.1. An open rosary^{1} R_{r} of length r is a two-pointed connected curve (R_{r}, p, q) such that

• R_{r} = L_{1} ∪_{a_{1}} L_{2} ∪_{a_{2}} ⋯ ∪_{a_{r−1}} L_{r} where L_{i} is a smooth rational curve,

i = 1, …, r;

• L_{i} and L_{i+1} meet each other in a single tacnode a_{i}, for i = 1, …, r − 1;

• L_{i} ∩ L_{j} = ∅ if |i − j| > 1;
• p ∈ L_{1} and q ∈ L_{r} are smooth points.

Remark 8.2. An open rosary of length r has arithmetic genus r − 1. Note that an open rosary of length r = 2r′ is naturally an open elliptic chain of length r′.

Definition 8.3. We say that a curve C admits an open rosary or length r if there is a 2-pointed open rosary (R_{r}, p, q) and a morphism ι : R_{r} ) C such that

• ι is an isomorphism onto its image over R_{r} \ {p, q};

• ι(p), ι(q) are nodes of C; we allow the case ι(p) = ι(q). A closed rosary C is a curve admitting ι : C′ ) C as above with the second condition replaced by

• ι(p) = ι(q) at a tacnode of C. A closed rosary with broken beads is a curve expressible as a union of open rosaries.

Remark 8.4. If C admits an open rosary of length r ≥ 2 then C admits a weak elliptic chain. If r is even then C admits an elliptic chain. Thus a closed rosary of even length is also a closed weak elliptic chain.

Proposition 8.5. Consider the closed rosaries of genus r+1. If the genus is even then there is a unique closed rosary C (of the given genus) and the automorphism group Aut(C) is finite. If the genus is odd then the closed rosaries depend on one modulus and the connected component of the identity Aut(C)^{∘} is isomorphic to 𝔾_{m}.

There is a unique open rosary (R, p, q) of length r. If Aut(R, p, q) denotes the automorphisms fixing p and q then

Aut(R, p, q)^{∘} ≃ 𝔾_{m}.

1This name was suggested to us by Jamie Song.

Figure 5. Closed rosary of genus six

It acts on tangent spaces of the endpoints with weights satisfying

wt_{𝔾_{m}} (T_{p}R) = (−1)^{r}wt_{𝔾_{m}} (T_{q}R).

Proof. Let C be a closed r-rosary obtained by gluing r smooth rational curves {[s_{i}, t_{i}]} so that

\frac{∂}{∂(s_{r}/t_{r})} = α_{r} \frac{∂}{∂(t_{1}/s_{1})} ; \frac{∂}{∂(s_{i}/t_{i})} = α_{i} \frac{∂}{∂(t_{i+1}/s_{i+1})} , i = 1, 2, …, r − 1.

Let C′ be another such rosary with the gluing data

\frac{∂}{∂(s′_{r}/t′_{r})} = α′_{r} \frac{∂}{∂(t′_{1}/s′_{1})} ; \frac{∂}{∂(s′_{i}/t′_{i})} = α′_{i} \frac{∂}{∂(t′_{i+1}/s′_{i+1})} , i = 1, 2, …, r − 1.

Consider the morphism f : C̃ ) C̃′ between the normalizations of C and C′ given by [s_{i}, t_{i}] ) [β_{i}s′_{i}, t′_{i}]. For f to descend to an isomorphism from C to C′, the following is necessary and sufficient:

df ( \frac{∂}{∂(s_{i} /t_{i} )} ) = \frac{∂}{β_{i} ∂(s′_{i}/t′_{i})} = \frac{α′_{i}}{β_{i}} \frac{∂}{∂(t′_{i+1} /s′_{i+1} )} = α_{i} β_{i+1} \frac{∂}{∂(t′_{i+1} /s′_{i+1} )} = df (α_{i} \frac{∂}{∂(t_{i+1} /s_{i+1} )} )

This gives rise to β_{i}β_{i+1} = α′_{i}/α_{i} and β_{r}β_{1} = α′_{r}/α_{r}. Solving for β_{i}, we get

β_{i} = \{ \begin{matrix}\frac{α′_{i}α_{i+1} α′_{i+2} ···α_{r}}{α_{i} α′_{i+1} α_{i+2} ···α′_{r}} β_{1}, if r − i is odd \\ \frac{α′_{i}α_{i+1} α′_{i+2} ···α′_{r}}{α_{i} α′_{i+1} α_{i+2} ···α_{r}} β_{1}^{−1}, if r − i is even \end{matrix}

When r is odd, there is no constraint and all r-rosaries are isomorphic. When r = 2k,

(β_{1}β_{2})(β_{3}β_{4}) ⋯ (β_{2k−1}β_{2k}) = (β_{2}β_{3})(β_{4}β_{5}) ⋯ (β_{2k}β_{1})

44 BRENDAN HASSETT AND DONGHOON HYEON

forces the condition

(8.1)  \frac{α′_{1}α′_{3} ⋯ α′_{2k−1}}{α_{1}α_{3} ⋯ α_{2k−1}} = \frac{α′_{2}α′_{4} ⋯ α′_{2k}}{α_{2}α_{4} ⋯ α_{2k}} .

This means that the 2k-rosaries are parametrized by

\frac{α_{1}α_{3} ⋯ α_{2k−1}}{α_{2}α_{4} ⋯ α_{2k}} ∈ 𝔾_{m}.

To describe the automorphisms we take C′ = C. When r is odd we get β_{i} = β_{i}^{−1} for each i which implies that Aut(C)^{∘} is trivial. When r = 2k we get a unique solution

β_{1} = β_{2}^{−1} = β_{3} = … = β_{2k}^{−1}

and thus Aut(C)^{∘} ≃ 𝔾_{m}.

The open rosary case entails exactly the same analysis, except that we omit the gluing datum

\frac{∂}{∂(s_{r}/t_{r})} = α_{r} \frac{∂}{∂(t_{1}/s_{1})}

associated with the end points. Thus we get a 𝔾_{m}-action regardless of the parity of r. Our assertion on the weights at the distinguished points p and q follows from the computation above of the action on tangent spaces. □

Definition 8.6. By breaking the ith bead of a rosary (open or closed), we mean replacing L_{i} with a union L′_{i} ∪ L′′_{i} of smooth rational curves meeting in a node such that L′_{i} meets L_{i−1} in a tacnode a_{i−1} and L′′_{i} meets L_{i+1} in a tacnode a_{i+1} (Figure 6).

Figure 6. Breaking a bead of a rosary

8.2. Classification of automorphisms.

Proposition 8.7. A c-semistable curve C of genus ≥ 4 has infinite automorphisms if and only if

(1) C admits an open rosary of length ≥ 2, or

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 45

(2) C is a closed rosary of odd genus (possibly with broken beads).

Proof. We have already seen in Proposition 8.5 that closed rosaries of odd genus have infinite automorphisms.

Let C be a c-semistable curve of genus g ≥ 4 that is not a closed rosary. For C to have infinitely many automorphisms, it must have a smooth rational component, say C_{1}. To satisfy the stability condition and still give rise to infinite automorphisms, C_{1} has to meet the rest of the curve in one node and a tacnode, or in two tacnodes. We examine each case below:

(1) C_{1} meets the rest in one node a_{0} and in a tacnode a_{1}: For the automorphisms of C_{1} to extend to automorphisms of C, the irreducible component C_{2}(≠ C_{1}) containing a_{1} must be a smooth rational component - this follows easily from that an automorphism of C lifts to an automorphism of its normalization. Also, C_{2} has to meet the rest of the curve in one point a_{2} other than a_{1} since otherwise C_{1} ∪ C_{2} would be an elliptic tail (or a_{1} = a_{0} and C is of genus two).

(2) C_{1} meets the rest in two tacnodes a_{0} and a_{1}: For the automorphisms to extend to C, the components C_{0} ≠ C_{1} containing a_{0} and C_{2} ≠ C_{1} containing a_{1} must be smooth rational curves. Hence C contains C_{0}∪C_{1}∪C_{2} which is a rosary of length three. Moreover, C_{0} and C_{2} do not intersect: If they do meet, say at a_{2}, then either C = C_{0} ∪ C_{1} ∪ C_{2} and the genus of C is of genus three (if a_{2} is a node) or C is a closed rosary if a_{2} is a tacnode.

Iterating, we eventually produce an open rosary ι : R_{r} ) C of length r ≥ 2 containing C_{1} as a bead. □

Corollary 8.8. An h-semistable curve C of genus ≥ 4 has infinite automorphisms if and only if

(1) C admits an open rosary of odd length ≥ 3, or

(2) C is a closed rosary of odd genus (possibly with broken beads).

Let C be a c-semistable curve and suppose D is a Deligne-Mumford stabilization of C. In other words, there exists a smoothing of C

ϖ : C ) T

such that D = lim_{t)t_{0}} C_{t} in the moduli space of stable curves. Here, a smoothing is a flat proper morphism to a smooth curve with distinguished point (T, t_{0}) such that ϖ^{−1}(t_{0}) = C and the generic fiber is smooth.

46 BRENDAN HASSETT AND DONGHOON HYEON

Our classification result (Proposition 8.7) has the following immediate consequence:

Corollary 8.9. Suppose C is a c-semistable curve with infinite automorphism group. Then the Deligne-Mumford stabilization and the pseudo-stabilization of C admit an elliptic bridge.

Indeed, C necessarily admits a tacnode which means that its stabilization contains a connected subcurve of genus one meeting the rest of the curve in two points.

9. Interpreting the flip via GIT

We will eventually give a complete description of the semistable and stable points of Chow_{g,2} and Hilb_{g,2}. For our immediate purpose, the following partial result will suffice:

Theorem 9.1. If C is c-stable, i.e., a pseudostable curve admitting no elliptic bridges, then Ch(C) ∈ Chow_{g,2}^{s}. Thus the Hilbert point [C]_{m} ∈ Hilb_{g,2}^{s,m} for m ≫ 0.

Proof. The GIT-stable loci Chow_{g,2}^{s} and Hilb_{g,2}^{s,m} , m ≫ 0, contain the nonsingular curves by [Mum77, 4.15]. (We discussed the relation between Chow and asymptotic Hilbert stability in Section 3.6.)

Recall that Proposition 6.1 guarantees that c-semistable curves admits bicanonical embeddings. In particular, this applies to pseudostable curves without elliptic bridges.

Suppose that C is a singular pseudostable curve without elliptic bridges. Assume that Ch(C) is not in Chow_{g,2}^{s}. If Ch(C) is strictly semistable then it is c-equivalent to a semistable curve C′ with infinite automorphism group. It follows that C is a pseudo-stabilization of C′, and we get a contradiction to Corollary 8.9. Suppose Ch(C) is unstable and let C′ denote a semistable replacement (see Theorem 4.5). By uniqueness of the pseudo-stabilization, C′ is not pseudostable but has C as its pseudo-stabilization. It follows that C′ has a tacnode. However, the pseudo-stabilization of such a curve necessarily contains an elliptic bridge. □

With our current partial understanding of the GIT of bicanonical curves, we are ready to prove Theorem 2.12. Our main task is to establish Isomorphisms (2.5) and (2.6). Proposition 2.9 established the existence of a birational contraction morphism Ψ : \overline{M}_{g}^{ps} ) \overline{M}_{g}(7/10). The first step here is to show that \overline{M}_{g}^{cs} and \overline{M}_{g}^{hs} are birational contractions of \overline{M}_{g} and small contractions of \overline{M}_{g}^{ps}. In particular, we may identify the divisor class groups of these GIT quotients with the divisor class group of \overline{M}_{g}^{ps} (which in turn is a subgroup of the divisor class group of \overline{M}_{g}). Furthermore, we obtain

Γ(\overline{M}_{g}^{ps}, n(K\overline{M} _{g}^{ps} + αδ^{ps})) ≃ Γ(\overline{M}_{g}^{hs}, n(K\overline{M} {hs \atop g} + αδ^{hs})) ≃ Γ(\overline{M}_{g}^{cs}, n(K\overline{M} _{g}^{cs} + αδ^{cs}))

and Lemma 2.8 gives

(9.1)  \begin{aligned}\overline{M}_{g}(7/10) & ≃ & Proj (⊕_{n≥0}Γ(n(K\overline{M} _{g}^{cs} + 7/10δ^{cs}))) \\ \overline{M}_{g}(7/10 − ϵ) & ≃ & Proj (⊕_{n≥0}Γ(n(K\overline{M} {hs \atop g} + (7/10 − ϵ)δ^{hs}))). \end{aligned}

The second step is to compute the induced polarizations of \overline{M}_{g}^{cs} and \overline{M}_{g}^{hs} in the divisor class group of \overline{M}_{g}^{ps}. This will show that K\overline{M} _{g}^{cs} + 7/10δ^{cs} (resp. K\overline{M} {hs \atop g} + (7/10 − ϵ)δ^{hs}) is ample on \overline{M}_{g}^{cs} (resp. \overline{M}_{g}^{hs}). Isomorphisms (2.5) and (2.6) then follow from (9.1).

To realize our GIT quotients as contractions of \overline{M}_{g}, we apply Theorem 5.4 in the bicanonical case. Consider the complement of the Deligne-Mumford stable curves V_{g,2}^{ss,} in the GIT-semistable locus Chow_{g,2}^{ss}; we must show this has codimension ≥ 2. Since ϖ(Hilb_{g,2}^{ss,m} ) ⊂ Chow_{g,2}^{ss} and ϖ|Hilb_{g,2}^{s,m} is an isomorphism where ϖ denotes the cycle class map from Hilb to Chow, the analogous statement for the Hilbert scheme follows immediately.

Proposition 7.1 implies Chow_{g,2}^{ss} \ V_{g,2}^{ss,} parametrizes

• pseudostable curves that are not Deligne-Mumford stable, i.e., those with cusps; and

• c-semistable curves with tacnodes. The cuspidal pseudostable curves have codimension two in moduli; the tacnodal curves have codimension three. Indeed, a generic tacnodal curve of genus g is determined by a two-pointed curve (C′, p, q) of genus g − 2 and an isomorphism T_{p}C′ ≃ T_{q}C′. We conclude there exist rational contractions F^{cs} : \overline{M}_{g} −−→ \overline{M}_{g}^{cs} and F^{hs} : \overline{M}_{g} −−→ \overline{M}_{g}^{hs}.

It remains to show that we have small contractions G^{cs} : \overline{M}_{g}^{ps} −−→ \overline{M}_{g}^{cs} and G^{hs} : \overline{M}_{g}^{ps} −−→ \overline{M}_{g}^{hs}. To achieve this, we must establish that Δ_{1} is the unique exceptional divisor of F^{cs} (resp. F^{hs}). The exceptional locus of F^{cs} (resp. F^{hs}) lies in the complement to the GIT-stable curves in the moduli space

\overline{M}_{g} \ U_{g,2}^{s,} (resp. \overline{M}_{g} \ U_{g,2}^{s,m} ).

Chow stable points are asymptotically Hilbert stable (cf. Proposition 3.13), i.e., U_{g,2}^{s,} ⊂ U_{g,2}^{s,m} when m ≫ 0. It suffices then to observe that δ_{1} is the unique divisorial component of \overline{M}_{g} \ U_{g,2}^{s,} , which is guaranteed by Theorem 9.1.

Theorem 5.4 gives moving divisors on \overline{M}_{g} inducing the contractions F^{cs} and F^{hs}. We apply Equation (5.3), which in our situation takes the form

r(2)λ_{m2} − r(2m)mλ_{2} = (m − 1)(g − 1)((20m − 3)λ − 2mδ) ∼ (10 − \frac{3}{2m})λ − δ;

this approaches 10λ − δ as m ) . Thus we have

(F^{hs})^{∗}L_{m} ∼ (10 − \frac{3}{2m})λ − δ (mod δ_{1}), m ≫ 0

and

(F^{cs})^{∗}L ∼ 10λ − δ (mod δ_{1}).

Using the identity

K_{\overline{M}_{g}} = 13λ − 2δ

we obtain (for m ≫ 0)

(F^{hs})^{∗}L_{m} ∼ K_{\overline{M}_{g}} + (7/10 − ϵ(m)))δ (mod δ_{1}), ϵ(m) = 39/(200m − 30)

and

(F^{cs})^{∗}L ∼ K_{\overline{M}_{g}} + 7/10δ (mod δ_{1}).

It follows then that

(G^{hs})^{∗}L_{m} ∼ K\overline{M} _{g}^{ps} + (7/10 − ϵ(m)))δ^{ps}

and

(G^{cs})^{∗}L ∼ K\overline{M} _{g}^{ps} + 7/10δ^{ps}.

The proof of Theorem 2.12 will be complete if we can show that Ψ^{+} is the flip of Ψ. More precisely, for small positive ϵ ∈ ℚ, Ψ^{+} is a small modification of \overline{M}_{g}^{ps} with K\overline{M} {hs \atop g} +(7/10−ϵ)δ^{hs} ample. Since \overline{M}_{g}^{cs} and \overline{M}_{g}^{hs} ar both small contractions

of \overline{M}_{g}^{ps}, Ψ^{+} is small as well. And the polarization we exhibited on \overline{M}_{g}^{hs} gives the desired positivity, which completes the proof of Theorem 2.12.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 49

10. Stability under one-parameter subgroups

In this section, we analyze whether c-semistable curves are GIT-semistable with respect to the one-parameter subgroups of their automorphism group. We shall also use deformation theory to classify the curves that belong to basins of attraction of such curves.

Our analysis will focus primarily on the Hilbert points. Indeed, Corollary 6.4 shows that we can recover the sign of the Hilbert-Mumford index of the Chow point from the indices of the Hilbert points. And in view of the cycle map ϖ : Hilb_{g,2} ) Chow_{g,2}, if [C]_{m} ∈ A_{ρ}([C^{⋆}]_{m}) for m ≫ 0 then Ch(C) ∈ A_{ρ}(Ch(C^{⋆})).

10.1. Stability analysis: Open rosaries.

Proposition 10.1. Let C = D ∪_{a_{0},a_{r+1}} R be a c-semistable curve of genus g consisting of a genus g − r − 1 curve D meeting the genus r curve R in two nodes a_{0} and a_{r+1} where

R := L_{1} ∪_{a_{1}} L_{2} ∪_{a_{2}} ⋯ ∪_{a_{r}} L_{r+1}

is a rosary of length r + 1, and D ∩ L_{1} = {a_{0}} and D ∩ L_{r+1} = {a_{r+1}}. There is a one-parameter subgroup ρ coming from the automorphisms of C of the rosary R such that for all m ≥ 2,

(1) μ([C]_{m}, ρ) = 0 if r is even;

(2) μ([C]_{m}, ρ) = −m + 1 if r is odd. In particular, C is Hilbert unstable if R is of even length and strictly semistable otherwise.

An application of Corollary 6.4 then yields:

Corollary 10.2. Let C and ρ be as in Proposition 10.1. Then C is Chow strictly semistable with respect to ρ and ρ^{−1}.

Proof. Upon restricting ω_{C} to D and each component of L, we get • ω_{C}|_{D} ≃ ω_{D}(a_{0} + a_{r+1});

• ω_{C}|_{L_{1}} ≃ ω_{L_{1}}(a_{0} + 2a_{1});
• ω_{C}|_{L_{r+1}} ≃ ω_{L_{r+1}} (a_{r+1} + 2a_{r});
• ω_{C}|_{L_{i}} ≃ ω_{L_{i}}(2a_{i−1} + 2a_{i}), 2 ≤ i ≤ r.

Hence we may choose coordinates x_{0}, …, x_{N}, N = 3g − 4, such that (1) L_{1} is parametrized by

[s_{1}, t_{1}] ) [s_{1}^{2}, s_{1}t_{1}, t_{1}^{2}, 0, …, 0];

(2) L_{r+1} is parametrized by

[s_{r+1}, t_{r+1}] ) [\underbrace{0, …, 0}_{3r−2}, s_{r+1}t_{r+1}, s_{r+1}^{2}, t_{r+1}^{2}, 0, …, 0].

(3) For 2 ≤ j ≤ r, L_{j} is parametrized by

[s_{j}, t_{j}] ) [\underbrace{0, …, 0}_{3j−5}, s_{j}^{3}t_{j}, s_{j}^{4}, s_{j}^{2}t_{j}^{2}, s_{j}t_{j}^{3}, t_{j}^{4}, 0, …, 0].

(4) D is contained in the linear subspace

x_{1} = x_{2} = ⋯ = x_{3r−1} = 0

and a_{0} = [1, 0, …, 0] and a_{r+1} = [\underbrace{0, …, 0}_{3r}, 1, 0, …, 0].

From the parametrization, we obtain a set of generators for the ideal of L:

(10.1)  \begin{matrix}x_{1}^{2} − x_{0}x_{2} − x_{2}x_{3}, x_{0}x_{3}, x_{0}x_{4}, …, x_{0}x_{3r}, \\ x_{i}x_{i+5}, x_{i}x_{i+6}, …, x_{i}x_{3r}, i = 1, 2, …, 3r − 5, \end{matrix}

and for j = 1, 2, …, r − 1, (10.2)

\begin{matrix}x_{3j−1}x_{3j+3}, x_{3j}x_{3j+3}, x_{3j}x_{3j+4}, x_{3j+1}^{2} − x_{3j}x_{3j+2} − x_{3j+2}x_{3j+3}, x_{3j}^{2} − x_{3j−1}x_{3j+2}, \\ x_{3j−2}x_{3j+1} − x_{3j−1}x_{3j+2}, x_{3j−2}x_{3j} − x_{3j−1}x_{3j+1}, x_{3j−2}x_{3j+2} − x_{3j}x_{3j+1}. \end{matrix}

In Proposition 8.5 we showed that 𝔾_{m} acts on the open rosary via automorphisms. With respect to our coordinates, this is the one-parameter subgroup ρ with weights:

(2, 1, 0, 2, 3, 4, 2, 1, 0, …, 2, 3, 4, \underbrace{2, 2, …, 2}_{N−3r}), if r is even

(2, 1, 0, 2, 3, 4, 2, 1, 0, …, 2, 1, 0, \overbrace{2, 2, …, 2}), if r is odd.

By considering the parametrization, it is easy to see that C is stable under the action of ρ.

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 51

Now we shall enumerate the degree two monomials in the initial ideal of C. From (10.1) and (10.2), we get the following monomials in x_{0}, …, x_{3r}:

(10.3)  \begin{matrix}x_{0}x_{2}, x_{0}x_{3}, x_{0}x_{4}, …, x_{0}x_{3r}, \\ x_{i}x_{i+5}, x_{i}x_{i+6}, …, x_{i}x_{3r}, i = 1, 2, …, 3r − 5, \\ x_{3j−1}x_{3j+3}, x_{3j}x_{3j+3}, x_{3j}x_{3j+4}, x_{3j}x_{3j+2}, \\ x_{3j−1}x_{3j+2}, x_{3j−2}x_{3j+1}, x_{3j−2}x_{3j}, x_{3j−2}x_{3j+2}, j = 1, 2, …, r − 1. \end{matrix}

The weights of these (9r^{2} − 5r)/2 monomials sum up to give

18r^{2} − 10r, if r is even;
18r^{2} − 19r + 7, if r is odd.

The total weight ∑_{i≤j, 0≤i,j≤3r} wt_{ρ}(x_{i}x_{j}) of all degree two monomials in x_{0}, …, x_{3r}

is

(3r + 2)(6r + 2), if r is even;
(3r + 2)(6r − 1), if r is odd.

Hence the degree two monomials in x_{0}, …, x_{3r} that are not in the initial ideal contributes, to the total weight,

(10.4)  \{ \begin{matrix}(3r + 2)(6r + 2) − (18r^{2} − 10r) = 28r + 4, if r is even; \\ (3r + 2)(6r − 1) − (18r^{2} − 19r + 7) = 28r − 9, if r is odd. \end{matrix}

The rest of the contribution comes from the monomials supported on the component D: These are the degree two monomials in x_{0}, x_{3r}, x_{3r+1}, …, x_{N} that vanish at a_{0} and a_{r+2}. The number of such monomials is, by Riemann-Roch,

(10.5)  h^{0}(D, O_{D}(2)(−a_{0} − a_{r+2})) = 7(g − r − 1) − 1.

Since wt_{ρ}(x_{i}) = 2 for all i = 0, 3r, 3r + 1, …, N, these monomials contribute 28g − 28r − 32 to the sum. Combining (10.4) and (10.5), we find the sum of the weights of the degree two monomials not in in(C) to be

28g − 28, if r is even;
28g − 41, if r is odd.

On the other hand, the average weight is

\frac{2 · P(2)}{N + 1} ∑_{i=0}^{N} wt_{ρ}(x_{i}) = \{ \begin{matrix}28g − 28, if r is even; \\ 28g − 42, if r is odd. \end{matrix}

Hence by (3.1), we find that μ([C]_{2}, ρ) = 0 if r is even and μ([C]_{2}, ρ) = −1 if r is odd.

We enumerate the degree three monomials in the same way: The degree three monomials in x_{0}, …, x_{3r} that are in the initial ideal are the multiples of (10.3) together with

(10.6)  x_{3j+1}^{2}x_{3j+3}, j = 0, 1, …, r − 1,

that come from the linear relation

x_{3j+2}(x_{3j}x_{3j+3}) − x_{3j+3}(x_{3j+1}^{2} − x_{3j}x_{3j+2} − x_{3j+2}x_{3j+3}) = 0.

From this, we find that the degree three monomials in x_{0}, …, x_{3r} that are not in the initial ideal contribute

66r + 6, if r is even;
66r − 25, if r is odd.

The contribution from D is

6h^{0}(D, O_{D}(3)(−a_{0} − a_{r+2})) = 6(11g − 11r − 12) = 66g − 66r − 72.

Hence the grand total is

∑_{j=1}^{P(3)} wt_{ρ}(x^{a(j)} ) = \{ \begin{matrix}66g − 66, if r is even; \\ 66g − 97, if r is odd \end{matrix}

On the other hand, the average weight is

\frac{3P(3)}{N + 1} ∑_{i=0}^{N} wt_{ρ}(x_{i}) = \{ \begin{matrix}66g − 66, if r is even; \\ 66g − 99, if r is odd. \end{matrix}

Using (3.1), we compute

μ([C]_{3}, ρ) = \{ \begin{matrix}0, if r is even; \\ −2, if r is odd. \end{matrix}

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 53

Corollary 6.4 implies

μ([C]_{m}, ρ) = \{ \begin{matrix}0, if r is even; \\ −m + 1, if r is odd. \end{matrix}

for each m ≥ 2. □

10.2. Basin of attraction: Open rosaries. Let C and R be as in the previous section. Let x_{i}, y_{i} be homogeneous coordinates on L_{i}. We may assume that

a_{0} = [0, 1]; a_{r+1} = [1, 0]; a_{i} = \{ \begin{matrix}[1, 0] = on L_{i} \\ [0, 1] = 0 on L_{i+1} \end{matrix}

Consider the 𝔾_{m} action associated to ρ. The action on R is given by (t, [x_{i}, y_{i}]) ) [x_{i}, t^{(−1)^{i−1}} y_{i}] on each L_{i}. Hence it induces an action on the tangent space T_{a_{i}} L_{i} given by

(t, \frac{∂}{∂(y_{i}/x_{i})} ) ) \frac{∂}{∂ (t^{(−1)^{i−1}} y_{i}/x_{i})} = t^{(−1)^{i}} \frac{∂}{∂(y_{i}/x_{i})} .

There is an induced 𝔾_{m} action on the Hilbert scheme and Hilb_{g,2}. Corollary 6.3 asserts that a neighborhood of [C] in the Hilbert scheme dominates the product of the versal deformation spaces. These inherit a 𝔾_{m} action as well, which we shall compute explicitly. (A) 𝔾_{m} action on the versal deformation spaces of nodes a_{0} and a_{r}: Let z be a local parameter at a_{0} on D. We have x_{1}/y_{1} as a local parameter at a_{0} on L_{0} and the local equation at a_{0} on C is z · (x_{1}/y_{1}) = 0. Hence the action on the node a_{0} is given by (z, x_{1}/y_{1}) ) (z, t^{−1}x_{1}/y_{1}) and the action on the versal deformation space is

c_{0} ) t^{−1}c_{0}.

Likewise, at a_{r+1}, the action on the node is

(y_{r+1}/x_{r+1}, z′) ) (t^{(−1)^{r+1}} y_{r+1}/x_{r+1}, z′)

where z′ is a local parameter at a_{r+1} on D, and the action on the versal deformation space is

c_{0} ) t^{(−1)^{r}} c_{0}.

(B) 𝔾_{m} action on the versal deformation space of a tacnode a_{i}: At a_{i}, the local analytic equation is of the form y^{2} = x^{4} where x := (y_{i}/x_{i}, x_{i+1}/y_{i+1}) and y := ((y_{i}/x_{i})^{2}, −(x_{i+1}/y_{i+1})^{2}) in k[[y_{i}/x_{i}]] ⊕ k[[x_{i+1}/y_{i+1}]] and the 𝔾_{m} action at the tacnode is given by

\begin{matrix}t.x = (t^{(−1)^{i−1}} y_{i}/x_{i}, x_{i+1}/(t^{(−1)^{i}} y_{i+1})) = t^{(−1)^{i−1}} x \\ t.y = t^{2(−1)^{i−1}} y. \end{matrix}

Therefore the action on the versal deformation space is

(c_{0}, c_{1}, c_{2}) ) (t^{4(−1)^{i−1}} c_{0}, t^{3(−1)^{i−1}} c_{1}, t^{2(−1)^{i−1}} c_{2}).

From these observations, we conclude that the basin of attraction of C with respect to ρ contains arbitrary smoothings of a_{2k+1} but no smoothing of a_{2k} for all 0 ≤ k < ⌈(r + 1)/2⌉. We have established:

Proposition 10.3. Retain the notation of Proposition 10.1 and assume that m ≫ 0.

(1) If r is even (i.e., the length of the rosary is odd) then A_{ρ}([C]_{m}) (resp. A_{ρ^{−1}} ([C]_{m})) parametrizes the curves consisting of D and a weak elliptic chain C′ of length r/2 meeting D in a node at a_{0} and in a tacnode at a_{r+1} (resp. in a tacnode at a_{0} and in a node at a_{r+1}) (Figure 7);

(2) If r is odd (i.e., the length of the rosary is even) then A_{ρ}([C]_{m}) (resp. A_{ρ^{−1}} ([C]_{m})) parametrizes the curves consisting of D and an elliptic chain C′ of length (r + 1)/2 (resp. length (r − 1)/2) meeting D in a node (resp. tacnode) at a_{0} and a_{r+1} (Figure 8). When r = 1, A_{ρ}([C]_{m}) consists of tacnodal curves normalized by D.

It follows from Proposition 10.3 and Proposition 10.1 that

Proposition 10.4. If a bicanonical curve admits an open elliptic chain then it is Hilbert unstable. In particular, a bicanonical curve with an elliptic bridge is Hilbert unstable.

The closed case can be found in Proposition 10.8 and Corollary 10.9.

10.3. Stability analysis: Closed rosaries.

Proposition 10.5. Let C be a bicanonical closed rosary of even length r. Then C is Hilbert strictly semistable with respect to the one-parameter subgroup ρ : 𝔾_{m} ) SL(3r) arising from Aut(C).

Figure 7. Basin of attraction of an open rosary of length five

Figure 8. Basin of attraction of an open rosary of length four

The relevant one-parameter subgroup was introduced in Proposition 8.5.

Proof. Restricting ω_{C}^{⊗2} to each component L_{i}, we find that each L_{i} is a smooth conic in ℙ^{3g−4}. We can choose coordinates x_{0}, …, x_{N} such that L_{i} is parametrized by

• [s_{i}, t_{i}] ) [\underbrace{0, …, 0}_{3(i−1)}, s_{i}^{3}t_{i}, s_{i}^{4}, s_{i}^{2}t_{i}^{2}, s_{i}t_{i}^{3}, t_{i}^{4}, 0, …, 0], i = 1, …, r − 1;

• [s_{r}, t_{r}] ) [s_{r}t_{r}^{3}, t_{r}^{4}, 0, …, 0, s_{r}^{3}t_{r}, s_{r}^{4}, s_{r}^{2}t_{r}^{2}]

56 BRENDAN HASSETT AND DONGHOON HYEON

The normalization of C admits the automorphisms given by

[s_{i}, t_{i}] ) [α^{sgn(i)} s_{i}, α^{1−sgn(i)} t_{i}], sgn(i) := i − 2⌊i/2⌋,

for i = 1, …, r − 1 and [s_{r}, t_{r}] ) [s_{r}, αt_{r}]. The one-parameter subgroup ρ associated to this automorphism has weights

(3, 4, 2, 1, 0, 2, ⋯, 3, 4, 2, 1, 0, 2)

The sum of the weights ∑ _{i=1}^{N} wt_{ρ}(x_{i}) is 6r if r is even and 6r + 3 if r is odd.

From the parametrization, we obtain a set of generators for the ideal of C: (10.7) x_{0}x_{5}, x_{0}x_{6}, …, x_{0}x_{3r−4}, x_{1}x_{5}, x_{1}x_{6}, …, x_{1}x_{3r−4}, x_{i}x_{i+5}, x_{i}x_{i+6}, …, x_{i}x_{3r−1}, i = 2, …, 3r − 6, x_{3j−2}x_{3j+2}, x_{3j−1}x_{3j+2}, x_{3j−1}x_{3j+3}, x_{3j}x_{3j+2}, x_{0}^{2} − x_{1}x_{2} − x_{1}x_{3r−1}, x_{3j}^{2} − x_{3j−1}x_{3j+1} − x_{3j+1}x_{3j+2}, x_{3r−1}^{2} − x_{1}x_{3r−2}, x_{3j−1}^{2} − x_{3j−2}x_{3j+1}, x_{0}x_{3r−3} − x_{1}x_{3r−2}, x_{3j−3}x_{3j} − x_{3j−2}x_{3j+1}, x_{0}x_{3r−1} − x_{1}x_{3r−3}, x_{0}x_{3r−2} − x_{3r−3}x_{3r−1}, x_{3j−3}x_{3j−1} − x_{3j−2}x_{3j}, j = 1, 2, …, r − 1, x_{3j}x_{3j+3}, x_{3j}x_{3j+4}, j = 1, 2, …, r − 2.

We have the following (9r^{2} − 11r)/2 degree two monomials that are in the initial ideal: (10.8) x_{0}^{2}, x_{0}x_{5}, x_{0}x_{6}, …, x_{0}x_{3r−1}, x_{1}x_{5}, x_{1}x_{6}, …, x_{1}x_{3r−4}, x_{1}x_{3r−2}, x_{i}x_{i+5}, x_{i}x_{i+6}, …, x_{i}x_{3r−1}, i = 2, …, 3r − 6, x_{3j−3}x_{3j−1}, x_{3j−3}x_{3j}, x_{3j−2}x_{3j+1}, x_{3j−2}x_{3j+2}, x_{3j−1}x_{3j+1}, x_{3j−1}x_{3j+2}, j = 1, 2, …, r − 1, x_{3j−1}x_{3j+3}, x_{3j}x_{3j+4}, j = 1, 2, …, r − 2.

The sum of the weights of these monomials is 34r − 18r^{2}. It follows that the sum of the weights of the monomials not in the initial ideal is

(3r − 1)6r − (34r − 18r^{2}) = 28r

which is precisely \frac{2P(2)}{N+1} ∑ _{i=0}^{N} wt_{ρ}(x_{i}). Hence μ([C]_{2}, ρ) = 0.

We shall now enumerate the degree three monomials in the initial ideal of C. Together with the monomials divisible by the monomials from (10.8), we have the initial terms

(10.9)  x_{3r−3}^{2}x_{3r−1}, x_{1}x_{3r−3}^{2}, x_{3j−2}x_{3j}^{2}, j = 1, 2, …, r − 1

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 57

that come from the Gröbner basis members

x_{1}x_{3r−3}^{2} − x_{3r−1}^{3}, x_{3r−3}^{2}x_{3r−1} − x_{3r−2}x_{3r−1}^{2}, x_{3j−2}x_{3j}^{2} − x_{3j−1}^{3}, j = 1, 2, …, r − 1.

The degree three monomials in (10.9) and the degree three monomials divisible by monomials in (10.8) have total weight 66r. This agrees with the average weight \frac{3P(3)}{N+1} ∑ _{i=0}^{N} wt_{ρ}(x_{i}) = \frac{3·11(g−1)}{3g−3} \frac{3g−3}{6} (3+4+2+1+0+2) = 66(g−1) = 66r. Therefore, μ([C]_{2}, ρ) = 0 = μ([C]_{3}, ρ) and C is m-Hilbert strictly semistable by Corollary 6.4.

Since Aut(C) ≃ 𝔾_{m}, a one-parameter subgroup coming from Aut(C) is of the form ρ^{a} for some a ∈ ℤ, and we have

μ([C]_{m}, ρ^{a}) = a μ([C]_{m}, ρ) = 0.
□

10.4. Basin of attraction: Closed rosaries.

Proposition 10.6. Retain the notation of Proposition 10.5. Then the basin of attraction A_{ρ}([C]_{m}) parametrizes the closed weak elliptic chains of length r/2 (Figure 9).

Proof. We use the parametrization from the proof of Proposition 10.5. C has

tacnodes a_{i} = [\underbrace{0, …, 0}_{3i+1}, 1, 0, …, 0], i = 1, …, k. From the parametrization, we

find that the local parameters x_{3i}/x_{3i+1} at a_{i} to the two branches is acted upon by ρ with weight (−1)^{i−1}. It follows that ρ acts on the versal deformation space (c_{0}, c_{1}, c_{2}) of the tacnode a_{i} with weights (4(−1)^{i−1}, 3(−1)^{i−1}, 2(−1)^{i−1}). Hence the basin of attraction A_{ρ}([C]) has arbitrary smoothings of a_{i} for odd i but no nontrivial deformations of a_{i} for even i.

□

10.5. Stability analysis: Closed rosaries with a broken bead. Closed rosaries with broken beads of even genus are unstable:

Proposition 10.7. Let r ≥ 3 be an odd number and C_{r} the curve obtained from a closed rosary of length r by breaking a bead. Then there exists a one-parameter subgroup ρ of Aut(C_{r}) with μ([C_{r}]_{m}, ρ) = 1 − m for each m ≥ 2, so [C_{r}]_{m} is unstable. Furthermore, Ch(C_{r}) is strictly semistable with respect to ρ.

58 BRENDAN HASSETT AND DONGHOON HYEON

Figure 9. Basin of attraction of a closed rosary of length six Figure 10. Closed rosary of genus six with a broken bead

Proof. Note that C_{r} is unique up to isomorphism and it can parametrized by

(10.10) \begin{matrix}• (s_{0}, t_{0}) & ) & (s_{0}t_{0}, s_{0}^{2}, t_{0}^{2}, 0, …, 0); \\ • (s_{1}, t_{1}) & ) & (0, 0, s_{1}^{2}, s_{1}t_{1}, t_{1}^{2}, 0, …, 0); \\ • (s_{i}, t_{i}) & ) & (\underbrace{0, …, 0}_{3(i−1)}, s_{i}^{3}t_{i}, s_{i}^{4}, s_{i}^{2}t_{i}^{2}, s_{i}t_{i}^{3}, t_{i}^{4}, 0, …, 0), i = 2, …, r − 1; \\ • (s_{r}, t_{r}) & ) & (s_{r}t_{r}^{3}, t_{r}^{4}, 0, …, 0, s_{r}^{3}t_{r}, s_{r}^{4}, s_{r}^{2}t_{r}^{2}). \end{matrix}

We give the set of monomials the graded ρ-weighted lexicographic order, where ρ is the one-parameter subgroup with the weight vector

(10.11)  (1, 0, 2, 1, 0, 2, 3, 4, 2, 1, 0, 2, 3, 4, 2, …, 1, 0, 2, 3, 4, 2).

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 59

A Gröbner basis for C_{r} is:

x_{0}x_{3}, x_{0}x_{4}, …, x_{0}x_{3r−4}; x_{1}x_{3}, x_{1}x_{4}, …, x_{1}x_{3r−4}; x_{2}x_{5}, x_{2}x_{6}, …, x_{2}x_{3r−1};

x_{0}x_{3r−2} − x_{3r−3}x_{3r−1}, x_{3r−1}^{2} − x_{1}x_{3r−2}, x_{3r−1}^{2} − x_{0}x_{3r−3},
x_{0}^{2} − x_{1}x_{2} − x_{1}x_{3r−1}, x_{0}x_{3r−1} − x_{1}x_{3r−3};
x_{6j+2}x_{6j+4} − x_{6j+3}^{2} + x_{6j+4}x_{6j+5}, j = 0, 1, …, \frac{r}{2} − 1;

and for j = 1, 2, …, r − 2,

x_{3j}x_{3j+2} − x_{3j+1}x_{3j+3}; x_{3j+2}^{2} − x_{3j+1}x_{3j+4}; x_{3j+2}^{2} − x_{3j}x_{3j+3};
x_{3j}x_{3j+4} − x_{3j+2}x_{3j+3}; x_{3j+2}x_{3j+4} − x_{3j+3}^{2} + x_{3j+4}x_{3j+5};
x_{3j}x_{3j+5}, x_{3j}x_{3j+6}, …, x_{3j}x_{3r−1}; x_{3j+1}x_{3j+5}, x_{3j}x_{3j+6}, …, x_{3j+1}x_{3j−1};
x_{3j+2}x_{3j+5}, x_{3j+2}x_{3j+6}, …, x_{3j+2}x_{3j−1},

together with the following degree three polynomials

(10.12)  \begin{matrix}x_{1}x_{3r−3}^{2} − x_{3r−1}^{3}, x_{3r−3}^{2}x_{3r−1} − x_{3r−2}x_{3r−1}^{2}, \\ x_{3j+1}x_{3j+3}^{2} − x_{3j+2}^{3}; j = 1, 2, …, r − 2. \end{matrix}

The degree two initial monomials are:

(10.13)  \begin{matrix}x_{0}^{2}, x_{0}x_{3}, x_{0}x_{4}, …, x_{0}x_{3r−1}; x_{1}x_{3}, x_{1}x_{4}, …, x_{1}x_{3r−4}, x_{1}x_{3r−2}; \\ x_{2}x_{4}, x_{2}x_{5}, x_{2}x_{6}, …, x_{2}x_{3r−1}; \end{matrix}

and for j = 1, 2, …, r − 2,

(10.14)  \begin{matrix}x_{3j}x_{3j+2}, x_{3j}x_{3j+3}, …, x_{3j}x_{3r−1}; x_{3j+1}x_{3j+4}, x_{3j}x_{3j+5}, …, x_{3j+1}x_{3r−1}; \\ x_{3j+2}x_{3j+4}, x_{3j+2}x_{3j+5}, …, x_{3j+2}x_{3r−1}. \end{matrix}

The sum of the weights of the monomials in (10.13) is 27r−33, whereas the monomials in (10.14) contribute 18r^{2} − 58r + 43 to the total weight of the monomials in the initial ideal.

The total weight of all degree two monomials is 18r^{2} − 3r − 3. Hence the weights of all degree two monomials not in the initial ideal sum up to

18r^{2} − 3r − 3 − (18r^{2} − 58r + 43) − (27r − 33) = 28r − 13

On the other hand, the average weight \frac{2P(2) ∑ r_{i}}{N+1} is 28r − 14. It follows from Proposition 3.7 that μ([C_{r}]_{2}, ρ) = (−(28r − 13) + 28r − 14) = −1.

The degree three monomials divisible by the ones in the lists (10.13), (10.14) contribute 27r^{3} + \frac{27}{2}r^{2} − \frac{159}{2}r^{2} + 26 to the total weight of the monomials in the initial ideal. On the other hand, the monomials

(10.15)  x_{1}x_{3r−3}^{2}, x_{3r−3}^{2}x_{3r−1}; x_{3j+2}^{3}, j = 1, 2, …, r − 2

Figure 11. Basin of attraction of a closed rosary with a broken bead

coming from the degree three Gröbner basis members (10.12) contribute 6r + 2. The sum of the weights of all degree three monomials is 27r^{3} + \frac{27}{2}r^{2} − \frac{15}{2}r − 3. Hence the total weight of the degree three monomials not in the initial ideal is

27r^{3} + \frac{27}{2}r^{2} − \frac{15}{2}r − 3 − (27r^{3} + \frac{27}{2}r^{2} − \frac{159}{2}r^{2} + 26) − (6r + 2) = 66r − 31.

On the other hand, the average weight is

\frac{3P(3)}{N + 1}(6r − 3) = 66r − 33.

By Proposition 3.7, the Hilbert-Mumford index is μ([C_{r}]_{3}, ρ) = −(66r − 31) + 66r−33) = −2. Since μ([C_{r}]_{2}, ρ) = 2μ([C_{r}]_{3}, ρ) < 0, it follows from Corollary 6.4 that C_{r} is m-Hilbert unstable for all m ≥ 2. Indeed, we find that

μ([C_{r}]_{m}, ρ) = 1 − m

for each m ≥ 2 and μ(Ch(C_{r}), ρ) = 0. □

10.6. Basin of attraction: Closed rosary with a broken bead.

Proposition 10.8. Let C_{r} and ρ be as in Proposition 10.7. Then the basin of attraction A_{ρ}([C]_{m}) parametrizes closed elliptic chains (C′, p, q) of length (r + 1)/2 such that ι(p) = ι(q) (Figure 11).

Corollary 10.9. A closed elliptic chain is Hilbert unstable.

Proof. At the node, the local analytic equation is given by

\frac{x_{0}}{x_{2}} · \frac{x_{3}}{x_{2}} = 0

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 61

and 𝔾_{m} acts on the local parameters x := x_{0}/x_{2} and y := x_{3}/x_{2} with weight −1. Hence 𝔾_{m} acts on the local versal deformation space (defined by xy = c_{0}) with weight −2. At the adjacent tacnode, 𝔾_{m} acts on the tangent space to the two branches with positive weights: The tangent lines are traced by x_{3}/x_{4} and x_{6}/x_{4}. In fact, 𝔾_{m} acts on the local versal deformation of the tacnode (defined by y^{2} = x^{4} + c_{2}x^{2} + c_{1}x + c_{0}) with a positive weight vector (2, 3, 4). Similar analysis reveals that 𝔾_{m} acts on the subsequent tacnode with a negative weight vector (−2, −3, −4). Using the symmetry of the rosary, we can conclude that 𝔾_{m} acts on the local versal deformation space of the tacnodes with weight vector alternating between (2, 3, 4) and (−2, −3, −4). The assertion now follows. □

11. Proofs of semistability and applications

Our main goal is a complete description of all c-equivalence and h-equivalence classes. Throughout, each c-semistable curve C is embedded bicanonically (cf. Proposition 6.1)

C ) ℙ^{3g−4},

and we consider the corresponding Chow points Ch(C) ∈ Chow_{g,2} and Hilbert points [C]_{m} ∈ Hilb_{g,2}, m ≫ 0. To summarize:

• If C is c-stable (resp. h-stable), then the equivalence class of C is trivial: It coincides with the SL_{3g−3} orbit of Ch(C) (resp. [C]_{m} for m ≫ 0).

• If C is strictly c- or h-semistable its equivalence class is nontrivial. We shall identify the unique closed orbit curve and describe all equivalent curves.

• Since closed orbit curves are separated in a good quotient [Ses72, 1.5], we have a complete classification of curves identified in the quotient spaces Hilb_{g,2}//SL_{3g−3} and Chow_{g,2}//SL_{3g−3}.

11.1. Elliptic bridges and their replacements.

Definition 11.1. An elliptic bridge of length k is a two-pointed curve (C′, p, q) (Figure 12) such that

• C′ = E_{1}∪_{a_{1}} ⋯∪_{a_{k−1}} E_{k} consists of connected genus-one curves E_{1}, …, E_{k}

such that E_{i} meets E_{i+1} in a node a_{i}, i = 1, 2, …, k − 1;
• E_{i} ∩ E_{j} ≠ ∅ if |i − j| ≠ 1;
• p ∈ E_{1} and q ∈ E_{k} are smooth points.

Figure 12. A generic elliptic bridge of length three

Figure 13. A generic closed elliptic bridge of length six and genus seven

An ordinary elliptic bridge is an elliptic bridge of length one.

Let C be a strictly c-semistable curve that is pseudostable, i.e., C has no tacnodes. Let E_{1}, …, E_{N} be the genus-one subcurves of C arising as components of elliptic bridges.

Lemma 11.2. Every c-semistable curve C′ admitting C as a pseudostable reduction can be obtained from the following procedure:

(1) Fix a subset

{E_{i}}_{i∈I} ⊂ {E_{1}, …, E_{N}}

of the genus-one subcurves arising in elliptic bridges.

(2) Choose a subset of the nodes of C lying on ∪_{i∈I}E_{i} consisting of points of

the following types:

• If E_{i} ∩ E_{i′} ≠ ∅ for some distinct i, i′ ∈ I then the node where they intersect must be included.

• Nodes where the E_{i}, i ∈ I meet other components may be included.

(3) Replace each of these nodes by a smooth ℙ^{1} (for any point of our subset) or by a chain of two smooth ℙ^{1}’s (only for points of the first type). Precisely,

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 63

let Z denote the curve obtained by normalizing our set of nodes and then joining each pairs of glued points with a ℙ^{1} or a chain of two ℙ^{1}’s with one component meeting each glued point.

(4) Let E′_{i} denote the proper transform of E_{i}, i ∈ I, which are pairwise disjoint in Z. Replace each E′_{i} with a tacnode. Precisely, write

D = Z \ ∪_{i∈I}E′_{i}

and consider a morphism

ν : Z ) C′

such that • ν|D is an isomorphism and ν|\overline{D} ) C′ is the normalization;

• for i ∈ I, ν contracts E′_{i} to a tacnode of C′. The generic curve C′ produced by this procedure does not admit components

isomorphic to ℙ^{1} containing a node of C′. We introduce ℙ^{1}’s in Step (3) only to separate two adjacent contracted elliptic components.

When enumerating the c-semistable curves, it is convenient to use a graph that is similar to the dual graph: We use a single line to denote a node and a double line to denote a tacnode (e.g. Figure 14).

Figure 14. A configuration corresponding to C_{1} − ℙ^{1} = ℙ^{1} = E − C_{2}

Example 11.3. Let C be the elliptic bridge of length one

C_{1} − E − C_{2}.

The possible Z are

C_{1} − E − C_{2}, C_{1} − ℙ^{1} − E − C_{2}, C_{1} − E − ℙ^{1} − C_{2}, C_{1} − ℙ^{1} − E − ℙ^{1} − C_{2}

64 BRENDAN HASSETT AND DONGHOON HYEON

and the possible C′ are

C_{1} − E − C_{2}, C_{1} = C_{2}, C_{1} − ℙ^{1} = C_{2}, C_{1} = ℙ^{1} − C_{2}, C_{1} − ℙ^{1} = ℙ^{1} − C_{2}.

The first two are the generic c-semistable configurations.

If C is an elliptic bridge of length two

C_{1} − E_{1} − E_{2} − C_{2}

then the possible Z are

C_{1} − E_{1} − E_{2} − C_{2}, C_{1} − ℙ^{1} − E_{1} − E_{2} − C_{2}, C_{1} − E_{1} − ℙ^{1} − E_{2} − C_{2}

C_{1} − E_{1} − E_{2} − ℙ^{1} − C_{2}, C_{1} − ℙ^{1} − E_{1} − ℙ^{1} − E_{2} − C_{2},
C_{1} − E_{1} − ℙ^{1} − ℙ^{1} − E_{2} − C_{2}, C_{1} − E_{1} − ℙ^{1} − E_{2} − ℙ^{1} − C_{2},

C_{1} − ℙ^{1} − E_{1} − ℙ^{1} − ℙ^{1} − E_{2} − C_{2}, C_{1} − ℙ^{1} − E_{1} − ℙ^{1} − E_{2} − ℙ^{1} − C_{2}, C_{1} − E_{1} − ℙ^{1} − ℙ^{1} − E_{2} − ℙ^{1} − C_{2}, C_{1} − ℙ^{1} − E_{1} − ℙ^{1} − ℙ^{1} − E_{2} − ℙ^{1} − C_{2}. The generic c-semistable configurations are

C_{1} − E_{1} − E_{2} − C_{2}, C_{1} = E_{2} − C_{2}, C_{1} − E_{1} = C_{2}, C_{1} = ℙ^{1} = C_{2}.

Proof. (of Lemma 11.2) Our hypotheses give a flat family

(†)  C′ ) B := Spec k[[t]]

whose generic fibre C′_{η} is smooth and the special fibre C′_{0} is C′. Furthermore, after a base change

\begin{matrix}B & ) & B_{1} = Spec k[[t_{1}]] \\ t & ) & t_{1}^{s} \end{matrix}

there exists a birational modification over B_{1}

ψ : C −−→ C′ ×_{B} B_{1}

such that C_{0} is C. In other words, C ) B_{1} is the pseudostable reduction of C′ ) B; we replace each tacnode by an elliptic bridge and contract any rational component that meets the rest of the curve in fewer than three points.

Let Z be the normalization of the graph of ψ, with π_{1} and π_{2} the projections to C and C′ ×_{B} B_{1} respectively:

\xymatrix{ & Z \ar[dl]_{π_{1}} \ar[dr]^{π_{2}} \\ C &  & C′ ×_{B} B_{1}}

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 65

By [Sch91, 4.4], Z is flat over B and Z = Z_{0} is reduced. An argument similar to [Sch91, 4.5-4.8] yields

• The exceptional locus of π_{2} is a disjoint union of connected genus-one

subcurves

⊔_{i∈I}E′_{i} ⊂ Z

that arise as proper transforms of components of elliptic bridges in C. Each component is mapped to a tacnode of C′.

• The exceptional locus of π_{1} is a union of chains of rational curves of length

one or two

⊔ℙ^{1} ⊔ (ℙ^{1} ∪ ℙ^{1}) ⊂ Z

that arise as proper transforms of rational components of C′ meeting the rest of the curve in two points (either two tacnodes or one node and one tacnode). Each component is mapped to a node of C contained in an elliptic bridge.

This yields the schematic description for the possible combinatorial types of C′.

We analyze the generic curves arising from our procedure. Suppose there is a component isomorphic to ℙ^{1} meeting the rest of the curve in a node and a tacnode. Corollary 6.3 implies we can smooth the node to get a c-semistable curve. The smoothed curve also arises from our procedure. □

Remark 11.4. Lemma 11.2 yields a bijection between subsets

{E_{i}}_{i∈I} ⊂ {E_{1}, …, E_{N}}

and generic configurations of the locus of curves arising from our procedure. Indeed, there is a unique generic configuration contracting the curves {E_{i}}_{i∈I}.

Proposition 11.5. Let C be strictly c-semistable without tacnodes and E_{1}, …, E_{N} the genus-one subcurves of C arising as components of elliptic bridges. Let C^{⋆} be the curve obtained from C by replacing each E_{i} with an open rosary (R_{i}, p_{i}, q_{i}) of length two. Then there exists a one-parameter subgroup

ρ : 𝔾_{m} ) Aut(C^{⋆})

such that Ch(C) ∈ A_{ρ}(Ch(C^{⋆})) and μ(Ch(C^{⋆}), ρ) = 0.

If C′ is another c-semistable curve with pseudostable reduction C then there exists a one-parameter subgroup

ρ′ : 𝔾_{m} ) Aut(C^{⋆})

such that Ch(C′) ∈ A_{ρ′}(Ch(C^{⋆})) and μ(Ch(C^{⋆}), ρ′) = 0.

Proof. The assmption that C is strictly c-semistable without tacnodes ensures it contains an elliptic chain of length one, i.e., an elliptic bridge.

The analysis of Proposition 8.5 makes clear that our description of C^{⋆} determines it uniquely up to isomorphism. Furthermore, we have

Aut(C^{⋆})^{∘} ≃ 𝔾_{m}^{N}

with basis {ρ_{1}, …, ρ_{N}}; here ρ_{i} denotes the one-parameter subgroup acting trivially on R_{j}, j ≠ i and with weight 1 on the tangent spaces T_{p_{i}} R_{i} and T_{q_{i}} R_{i}. As explained in § 10.2, it acts with negative weights on the versal deformation space of the tacnode of R_{i}.

Consider the one-parameter subgroup

ρ = ∏_{i=1}^{N} ρ_{i}^{−1},

which acts with positive weights on each of the tacnodes. The basin of attraction analysis of Proposition 10.3 shows that A_{ρ}(Ch(C)) parametrizes those curves obtained by replacing each open rosary of C^{⋆} with an elliptic bridge/chain of length one. This includes our original curve C.

Now for any one-parameter subgroup

ρ′ = ∏_{i=1}^{N} ρ_{i}^{−e_{i}} ,

we can compute

μ(Ch(C^{⋆}), ρ′) = − ∑_{i=1}^{N} e_{i} μ(Ch(C^{⋆}), ρ_{i}) = 0

using Corollary 10.2. In particular, we have

μ(Ch(C^{⋆}), ρ) = 0.

Section 10.2 gives the action of ρ′ on the versal deformations of the singularities of C^{⋆}. It acts with weights (2e_{i}, 3e_{i}, 4e_{i}) on the versal deformation space of the tacnode on R_{i}. At a node (p_{i} or q_{i}) lying on a single open rosary R_{i} of length two, it acts with weight −e_{i}. For nodes on two open rosaries R_{i} and R_{j}, it acts with weight −(e_{i} + e_{j}).

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 67

Restrict attention to one-parameter subgroups with weights e_{i} ≠ 0 for each i, j = 1, …, N. These naturally divide up into 2^{N} equivalence classes, depending on the signs of the e_{i}. Let I ⊂ {1, …, N} denote those indices with e_{i} < 0. Just as in the proof of Proposition 10.3, the basin of attraction A_{ρ′}(Ch(C^{⋆})) does not contain smoothings of the tacnodes in R_{i}, i ∈ I but does contain all smoothings of the remaining tacnodes. Choosing the negative e_{i} suitably large in absolute value, we can assume each −(e_{i} + e_{j}) > 0, so the nodes where two rosaries meet are smoothed provided at least one of the adjacent tacnodes is not smoothed.

Thus A_{ρ′}(Ch(C^{⋆})) consists of the c-semistable curves obtained by smoothing all the tacnodes not indexed by I, as well as the nodes on the rosaries containing one of the remaining tacnodes (indexed by I). The generic member of the basin equals the generic configuration indexed by I, as described in Remark 11.4. It follows that each curve C′ enumerated in Lemma 11.2 appears in the the basin of attraction of Ch(C^{⋆}) for a suitable one-parameter subgroup ρ′. □

11.2. Chow semistability of c-semistable curves. Here we prove that bicanonical c-semistable curves are Chow semistable. By Theorem 9.1, it suffices to consider curves that are not c-stable.

Let C′ denote a strictly c-semistable curve, with tacnodes and/or elliptic bridges. Assume that C′ is Chow unstable and let

C′ ) B := Spec k[[t]]

be a smoothing. Let C′′ be a Chow semistable reduction of this family (see Theorem 4.5) and C the pseudostable reduction.

Reversing the steps outlined in Lemma 11.2, we see that C is obtained by replacing each tacnode of C′ (or C′′) with an elliptic bridge and then pseudostabilizing. Let C^{⋆} denote the curve obtained from C in Proposition 11.5, which guarantees that Ch(C′) and Ch(C′′) are contained in basins of attraction A_{ρ′}(Ch(C^{⋆})) and A_{ρ′′}(Ch(C^{⋆})) respectively. Moreover, since

μ(Ch(C^{⋆}), ρ′) = μ(Ch(C^{⋆}), ρ′′) = 0

Lemma 4.3 implies that C′ (resp. C′′) is Chow semistable iff C is Chow semistable. This contradicts our assumption that C′ is Chow unstable.

Next, we give a characterizaztion of the closed orbit curves in c-equivalence classes of strictly semistable curves:

68 BRENDAN HASSETT AND DONGHOON HYEON

Proposition 11.6. A strictly c-semistable curve has a closed orbit if and only if • each tacnode is contained in an open rosary;

• each open rosary has length two; and

• there are no elliptic bridges other than length two rosaries.

Since each length-two rosary has one tacnode and contributes a 𝔾_{m}-factor to Aut(C), we have:

Corollary 11.7. If C is a strictly c-semistable curve with closed orbit then

Aut(C)^{∘} ≃ 𝔾_{m}^{τ}.

where τ is the number of tacnodes and the superscript ∘ denotes the connected component of the identity.

Proof of Proposition 11.6. Assume that C′ is a strictly semistable curve with closed orbit. Let C be a pseudostable reduction and C^{⋆} the curve specified in Proposition 11.5, so the Chow point of C′ is in the basin of attraction of the Chow point of C^{⋆}. Since C^{⋆} is Chow semistable, we conclude that C′ = C^{⋆}.

Conversely, suppose C′ is a curve satisfying the three conditions of Proposition 11.6. Again, let C be a pseudostable reduction of C′ and C^{⋆} the curve obtained in Proposition 11.5, so that C′ is in the basin of attraction of C^{⋆} for some one-parameter subgroup ρ′. Note that C^{⋆} also satisfies the conditions of Proposition 11.6. The basin of attraction analysis in Section 10.2 implies that any nontrivial deformation of C^{⋆} in A_{ρ′}(Ch(C^{⋆})) induces a nontrivial deformation of at least one of the singularities of C^{⋆} sitting in an open rosary.

There are three cases to consider: First, we could deform the tacnode on one of the rosaries R_{i}. However, then the rosary R_{i} deforms to an elliptic bridge in C′ that is not a length two rosary, which yields a contradiction. Therefore, we may assume that none of the tacnodes in C^{⋆} is deformed in C′. Second, we could smooth a node where length two rosaries meet. However, this would yield a rosary in C′ of length > 2. Finally, we could smooth a node where a length two rosary R_{i} meets a component not contained in an rosary. However, the tacnode of R_{i} then deforms to a tacnode of C′ not on any length two rosary. □

11.3. Hilbert semistability of h-semistable curves. Suppose that C is a hsemistable bicanoncal curve. By definition it is also c-semistable and thus Chowsemistable by the analysis of Section 11.2. Of course, strictly Chow-semistable

Figure 15. Degeneration to the c-semistable closed orbit curve

points can be Hilbert unstable, and we classify these in two steps. First, we enumerate the curves C_{0} with strictly semistable Chow point such that there exists a one-parameter subgroup ρ : 𝔾_{m} ) Aut(C_{0}) destabilizing the Hilbert point of C_{0}, i.e., with μ([C_{0}]_{m}, ρ) < 0 for m ≫ 0. Second, we list the curves that are in the basins of attraction A_{ρ}([C_{0}]_{m}), which are also guaranteed to be Hilbert unstable by the Hilbert-Mumford one-parameter subgroup criterion. Proposition 4.4 shows these are all the Hilbert unstable curves: If C is a c-semistable bicanonical curve that is Hilbert unstable then its Chow point is contained in the basin of attraction A_{ρ}(Ch(C_{0})) of a Chow semistable curve C_{0} with closed orbit such that μ([C_{0}]_{m}, ρ) < 0.

If the genus is odd and C_{0} is a closed rosary (without broken beads) then C_{0} is Hilbert semistable with respect to any 1-PS coming from Aut(C_{0}) (Proposition 10.5).

Suppose that C_{0} has open rosaries S_{1}, …, S_{ℓ}. Each contributes 𝔾_{m} to the automorphism group of C_{0} and Aut(C_{0})^{∘} ≃ 𝔾_{m}^{×ℓ}. Let p_{i}, q_{i} denote the nodes in the intersection S_{i} ∩ \overline{C_{0} − S_{i}}. The automorphism coming from S_{i} gives rise to a

70 BRENDAN HASSETT AND DONGHOON HYEON

one-parameter subgroup

ρ_{i} : 𝔾_{m} −)^{≃} \{1\} × ⋯ × \underbrace{𝔾_{m}}_{ith}× ⋯ × \{1\} ) 𝔾_{m}^{×ℓ} ≃ G_{Ch(C_{0})}^{∘}

where the second 𝔾_{m} means the ith copy in the product 𝔾_{m}^{×ℓ} and G_{Ch(C_{0})} is the stabilizer group. We assume that S_{1}, …, S_{k} are open rosaries of even length and S_{k+1}, …, S_{ℓ} are of odd length. For i ≤ k, the weights of ρ_{i} on the versal deformation spaces of p_{i} and q_{i} have the same sign (see §10.2). We normalize ρ_{i} so that this weight is negative.

Given our one-parameter subgroups ρ : 𝔾_{m} ) Aut(C_{0})^{∘} with μ([C_{0}]_{m}, ρ) < 0, we can expand

ρ = ∏_{i=1}^{k} ρ_{i}^{a_{i}} × ∏_{i=k+1}^{ℓ} ρ_{i}^{b_{i}} , a_{i}, b_{i} ∈ ℤ

so that

μ([C_{0}]_{m}, ρ) = ∑_{i=1}^{k} a_{i} μ([C_{0}]_{m}, ρ_{i}) + ∑_{i=k+1}^{ℓ} b_{i} μ([C_{0}]_{m}, ρ_{i}) < 0.

We have already computed these terms: Proposition 10.1 implies that μ([C_{0}]_{m}, ρ_{i}) = 0 for i = k + 1, …, ℓ and μ([C_{0}]_{m}, ρ_{i}) = 1 − m for i = 1, …, k. Thus in order for the sum to be negative, we must have a_{i} > 0 for some i = 1, …, k. In particular, there is at least one rosary of even length. Proposition 10.3 implies that the basin of attraction A_{ρ}([C_{0}]_{m}) contains curves with elliptic chains, which are not h-semistable.

We are left with the case of a closed rosary C_{r} of even genus with one broken bead. There is a unique one-parameter subgroup ρ of the automorphism group, and we choose the sign so that it destabilizes C_{r} (cf. Proposition 10.7). The basin of attraction analysis in Proposition 10.8 again shows that the curves with unstable Hilbert points admit elliptic chains.

Thus curves with unstable Hilbert points are not h-semistable, which completes our proof that h-semistable curves are Hilbert semistable.

We shall now prove that if C is h-stable then it is Hilbert stable. If C is Hilbert strictly semistable, then it belongs to a basin of attraction A_{ρ}([C_{0}]_{m}) where C_{0} is a Hilbert semistable curve with infinite automorphisms and ρ is a 1-PS coming from Aut(C_{0}). By Corollary 8.9, C_{0} admits an open rosary of odd length ≥ 3 or

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 71

is a closed rosary of even length ≥ 4. But we showed in Propositions 10.3 and 10.6 that any curve in the basin of such C_{0} has a weak elliptic chain and hence is not h-stable.

Finally, we characterize the closed orbits of strictly h-semistable curves. These do not admit elliptic chains, and in particular, do not admit open rosaries of even length (see Remark 8.4).

Proposition 11.8. A strictly h-semistable curve has a closed orbit if and only if • it is a closed rosary of odd genus; or

• each weak elliptic chain is contained in a chain of open rosaries of length three.

Since each length three open rosary has two tacnodes and contributes 𝔾_{m} many automorphisms to Aut(C),

Corollary 11.9. If C is a strictly h-semistable curve with closed orbit then

Aut(C)^{∘} ≃ 𝔾_{m}^{τ/2}

where τ is the number of tacnodes.

Proof of Proposition 11.8. Suppose C′ is strictly h-semistable. We shall show that there exists a curve C^{∗} satisfying the conditions of Proposition 11.8 and a one-parameter subgroup ρ′ of Aut(C^{∗}) such that [C′]_{m} ∈ A_{ρ′}([C^{∗}]_{m}) for m ≫ 0 and μ([C^{∗}]_{m}, ρ′) = 0.

Assume first that C′ is a closed weak elliptic chain with r components, with arithmetic genus 2r + 1. Let C^{∗} denote a closed rosary with beads L_{1}, …, L_{2r} and tacnodes a_{1}, …, a_{2r}. Proposition 8.5 implies Aut^{∘}(C^{∗}) ≃ 𝔾_{m}, generated by a one-parameter subgroup ρ acting on the versal deformation spaces of the a_{2j} with positive weights and the a_{2j−1} with negative weights. Proposition 10.6 implies A_{ρ}([C^{∗}]_{m}) contains the closed weak elliptic chains of length r.

Now assume that C′ is not a closed weak elliptic chain but contains maximal closed weak elliptic chains C′′_{1}, …, C′′_{s} of lengths ℓ_{1}, …, ℓ_{s}. Let p_{j} (resp. q_{j}) denote the node (resp. tacnode) where C′′_{j} meets the rest of the curve. Let C^{∗} be the curve obtained from C′ by replacing each C′′_{j} with a chain of ℓ_{j} open rosaries of length three. Precisely, write

D = C′ \ (⋃_{j=1}^{s}C′′_{j} \ {p_{j}, q_{j}})

72 BRENDAN HASSETT AND DONGHOON HYEON

and let S_{j}, j = 1, …, s denote a chain of ℓ_{j} open rosaries of length three joined end-to-end. Then C^{∗} is obtained by gluing S_{j} to D via nodes at p_{j} and q_{j}. One special case requires further explanation: If C′ admits an irreducible component ≃ ℙ^{1} meeting the rest of C′ at two points q_{i} and q_{j} then we contract this component in C^{∗}.

Example 11.10. There are examples where the construction of C^{∗} involves components being contracted. Let C_{1} and C_{2} be smooth and connected of genus ≥ 2 and let E_{1} and E_{2} be elliptic. Consider the curve C′

C_{1} −_{p_{1}} E_{1} =_{q_{1}} ℙ^{1} =_{q_{2}} E_{2} −_{p_{2}} C_{2}.

Replacing the weak elliptic chains with rosaries of length three yields

C_{1} −_{p_{1}} ℙ^{1} = ℙ^{1} = ℙ^{1} −_{q_{1}} ℙ^{1} −_{q_{2}} ℙ^{1} = ℙ^{1} = ℙ^{1} −_{p_{2}} C_{2}

which is not h-semistable. Contracting the middle ℙ^{1}, we obtain C^{∗}:

C_{1} −_{p_{1}} ℙ^{1} = ℙ^{1} = ℙ^{1} − ℙ^{1} = ℙ^{1} = ℙ^{1} −_{p_{2}} C_{2}.

There are examples where D fails to be pure-dimensional. Start with the curve C′

C_{1} =_{q_{1}} E_{1} −_{p_{1}=p_{2}} E_{2} =_{q_{2}} C_{2}

with the C_{i} and E_{i} as above. Then C^{∗} is equal to

C_{1} −_{q_{1}} ℙ^{1} = ℙ^{1} = ℙ^{1} −_{p_{1}=p_{2}} ℙ^{1} = ℙ^{1} = ℙ^{1} −_{q_{2}} C_{2}.

We return to our proof: The curve C^{∗} has

Aut(C^{∗})^{∘} ≃ 𝔾_{m}^{N}, N = ∑_{j=1}^{s} ℓ_{j}.

Essentially repeating the argument of Proposition 11.5, using the one-parameter subgroup analysis of Proposition 10.1 and the basin-of-attraction analysis of Proposition 10.3 (or Proposition 10.7 and 10.8 in the generate case), we obtain a one-parameter subgroup ρ′ in the automorphism group such that [C′]_{m} ∈ A_{ρ′}([C^{∗}]_{m}) for m ≫ 0 and μ([C^{∗}]_{m}, ρ′) = 0.

We now show that the curves enumerated in Proposition 11.8 all have closed orbits. Due to [Kem78, Theorem 1.4], it suffices to show none of these are contained in the basin of attraction of any other. Suppose that C_{1}^{∗} and C_{2}^{∗} are such that

[C_{2}^{∗}]_{m} ∈ A_{ρ}([C_{1}^{∗}]_{m})

LOG MINIMAL MODEL PROGRAM FOR \overline{M}_{g}: FIRST FLIP 73

for some one-parameter subgroup ρ of Aut(C_{1}^{∗})^{∘}. A nontrivial deformation of C_{1}^{∗} necessarily deforms one of the singularities of C_{1}^{∗}. If the singularity is a tacnode on a length-three open rosary, the resulting deformation admits a weak elliptic chain that is not contained in a chain of length-three rosaries. If the singularity is a node where two length-three open rosaries meet then the deformation admits a weak elliptic chain not contained in a chain of length-three open rosaries.

However, there is one case that requires special care: Suppose that C_{1}^{∗} is a closed chain of r rosaries of length three

R_{1} −_{p_{12}} R_{2} −_{p_{23}} ⋯ −_{p_{r−2 r−1}} R_{r−1} −_{p_{r−1r}} R_{r} −_{p_{r1}} R_{1}

where R_{1} and R_{r} meet at a node p_{r1}; this has arithmetic genus 2r + 1. Let C_{2}^{∗} denote a closed rosary of genus 2r + 1, which is a deformation of C_{1}^{∗}. We need to insure that

(11.1)  [C_{2}^{∗}]_{m} ∉ A_{ρ}([C_{1}^{∗}]_{m})

for any one-parameter subgroup ρ of Aut([C_{1}^{∗}]_{m})^{∘}. We can express

ρ = ∏_{j=1}^{r} ρ_{j}^{e_{j}}

where ρ_{j} acts trivially except on R_{j} and has weights +1 and −1 on T_{p_{j−1j}} R_{j} and T_{p_{jj+1}} R_{j}. (Here ρ_{r} acts with weights +1 and −1 on T_{p_{r−1r}} R_{r} and T_{p_{r,1}} R_{r}.) However, assuming ρ is nontrivial, one of the following differences

e_{1} − e_{2}, …, e_{r} − e_{1}

is necessarily negative; for simplicity, assume e_{1} − e_{2} < 0. It follows that ρ acts with negative weight on the versal deformation of the node p_{12}, thus deformations in A_{ρ}([C_{1}^{∗}]_{m}) cannot smooth p_{12}. We conclude that deformations in the basin of attraction of C_{1}^{∗} cannot smooth each node, which yields (11.1) □

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