Integral points and effective cones of moduli spaces of stable maps

Brendan Hassett ^{∗}and Yuri Tschinkel ^{†}

May 8, 2002

Abstract

Consider the Fulton-MacPherson configuration space of n points on ℙ^{1}, which is isomorphic to a certain moduli space of stable maps to ℙ^{1}. We compute the cone of effective 𝔖_{n}-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of SL_{2} in the space of binary forms of degree n ≥ 3.

Contents

0 Introduction 2

1 Generalities 4

2 Construction of resolutions 8

3 Verification of exponents 15

4 The invariant effective cone 18

5 Final remarks 23

∗Partially supported by NSF grant 0196187 †Partially supported by NSF grant 0100277 and the Clay Foundation

0 Introduction

In this paper, we compute the 𝔖_{n}-invariant cone of effective divisors of the Fulton-MacPherson configuration space of n points on ℙ^{1}. This space is isomorphic to the moduli space \overline{M}_{0,n}(ℙ^{1}, 1) of stable maps of degree one from genus zero curves with n marked points to ℙ^{1}. We also compute the effective cone of the generic fiber of the natural map

\overline{M}_{0,n}(ℙ^{1}, 1)/𝔖_{n} → \overline{M}_{0,n}/𝔖_{n}.

Our motivation is to provide a geometric explanation of a formula, obtained by Duke, Rudnick and Sarnak, giving the asymptotic behavior of the number of binary forms of degree n with fixed discriminant and bounded integral coefficients. This fits into a larger program to predict and prove asymptotic formulas for the number of rational and integral points of bounded height on algebraic varieties.

We introduce a counting function for integral points on an algebraic variety as follows: given a variety U over a ring of integers 𝔬 and functions g_{1}, ..., g_{n}, regular on U, define

N(U, B) := {x ∈ U(𝔬) | max_{j}(∥g_{j}(x)∥) ≤ B},

where ∥ · ∥ is a valuation on 𝔬. This is finite only when the functions g_{j} give an embedding of U.

It is most natural to interpret the functions g_{j} as sections of a line bundle L on a projective compactification X ⊃ U defined over the fraction field F of 𝔬. The fact that the sections embed U implies that L is big, i.e., is contained in the interior of the effective cone Λ_{eff}(X) of X. Therefore, in order to describe all natural counting functions on open subsets of X we need to compute its effective cone. Furthermore, in many cases it can be proved that the asymptotic properties of N(U, B) are intimately related to the structure of this cone.

Let P (x) = P (x_{0}, ..., x_{r}) be a homogeneous polynomial of degree n in r+1 variables. A standard heuristic in number theory predicts that the number

N_{P} (B) := {x | max(|x_{j}|) ≤ B, P (x) = 0, and x ∈ ℤ^{r+1}}

of integral solutions of the equation P (x) = 0 of “height” ≤ B grows asymptotically like B^{r+1−n} as B → ∞. When the number of variables is ≫ 2^{n}, the affine variety V_{P} defined by P = 0 is smooth and there are no local obstructions, an asymptotic formula can be established using the classical circle method in analytic number theory (see [3], [25] and the references therein). Of course, there may be difficulties when the number of variables is small or the variety V_{P} is singular.

The following example appeared in the paper by Duke, Rudnick and Sarnak [6]. Consider the vector space of binary forms of degree n

x_{n}z^{n} + x_{n−1}z^{n−1}w + … + x_{0}w^{n}.

The algebraic group SL_{2} acts on this space by coordinate substitutions. When n = 3, the discriminant form

disc(x_{0}, ..., x_{3}) := 27x_{0}^{2}x_{3}^{2} − 18x_{0}x_{1}x_{2}x_{3} + 4x_{0}x_{2}^{3} + 4x_{1}^{3}x_{3} − x_{1}^{2}x_{2}^{2}

generates the ring of SL_{2}-invariants. Then there exists a constant c > 0 so that

N_{disc−1}(B) = cB^{2/3}(1 + o(1))

as B → ∞. Note that the exponent 2/3 is larger than what is predicted by the standard heuristic.

More generally, one has the

Theorem 0.1 [6] Fix a generic binary form f of degree n ≥ 3 with integral coefficients. Let N(B) be the number of binary forms SL_{2}(ℤ)-equivalent to f with coefficients bounded by B. Then there exists a c > 0 such that

N(B) = cB^{2/n}(1 + o(1)),

as B → ∞.

We give a geometric interpretation of the exponent 2/n in Theorem 0.1. To this end, we refine the heuristics for counting integral points to take into account singularities of the relevant varieties (see Conjecture 1.6). We verify that Conjecture 1.6 is consistent with Theorem 0.1 in Theorem 2.1. Its proof involves the computations of effective cones alluded to above. Acknowledgments: We are grateful to the Alfréd Rényi Institute of the Hungarian Academy of Sciences for organizing the conference at which much of this work was done.

1 Generalities

1.1 Singularities of pairs and effective cones

We work over a field of characteristic zero. Let X be a normal projective variety with canonical class K_{X} and let D be a reduced effective Weil divisor of X.

Definition 1.1 A good pair (X, D) consists of a smooth projective variety X and a strict normal crossings divisor D in X. This means that all irreducible components of D are smooth and intersect transversally.

Let (X, D) be a good pair and let Λ_{eff}(X) denote the closed cone of effective divisors classes of X; a divisor is big exactly when its class is in the interior of this cone. Define

a(L, D) := inf{a ∈ ℝ | aL + (K_{X} + D) ∈ Λ_{eff}(X)},

where we identify line bundles and their divisor classes. Note that a(L, D) is a positive real number whenever −(K_{X} + D) is big. The constant −a(L, D) is called the log-Kodaira energy of L (see [10]).

If (X, D) is not good then resolution of singularities implies the existence of a good resolution ρ : (X̃, D̃) → (X, D). Precisely, (X̃, D̃) is a good pair, ρ a birational projective morphism, and D̃ is the union of the exceptional divisors of ρ and the proper transform of D. Recall that (X, D) is logcanonical if K_{X} + D is ℚ-Cartier and

KX̃ + D̃ ≡_{ℚ} ρ^{∗}(K_{X} + D) + ∑ d_{j}E_{j},

where the E_{j} are the exceptional divisors of ρ and d_{j} ≥ 0 for all j.

Example 1.2 When X is a smooth surface, (X, D) is log-canonical only when the curve D is smooth or nodal. If X is smooth of arbitrary dimension, D must have at worse nodes in codimension one.

If L is a line bundle on X put

a(L, D) := a(ρ^{∗}L, D^{t}),

where D^{t} ⊂ X̃ is the total transform of D. Note that a(L, D) is computed on X̃.

Proposition 1.3 Let (X, D) be a log-canonical pair and assume that X −D has canonical singularities. If L is a big line bundle on X then

a(L, D) = inf{a ∈ ℝ | aL + (K_{X} + D) ∈ Λ_{eff}(X)}.

In particular, a(L, D) does not depend on the choice of a desingularization.

Proof. Choose a good resolution ρ : (X̃, D̃) → (X, D), so that

KX̃ + D̃ − ∑ d_{j}E_{j} = ρ^{∗}(K_{X} + D)

where d_{j} ≥ 0, and d_{j} ≥ 1 if ρ(E_{j}) ⊄ D. In particular, each exceptional divisor not contained in the total transform D^{t} has log discrepancy ≥ 1. Therefore, we have

αρ^{∗}(L) + KX̃ + D^{t} − ∑ d′_{j}E_{j} = ρ^{∗}(K_{X} + D + αL),

with each d′_{j} ≥ 0. For any ℚ-Cartier divisor M on X and effective divisor ∑ d′_{j}E_{j} supported in the exceptional locus of ρ, M is effective iff ρ^{∗}(M) + ∑ d′_{j}E_{j} is effective. □

Proposition 1.4 Let (X_{1}, D_{1}) and (X_{2}, D_{2}) be log-canonical pairs, so that X_{1}−D_{1} and X_{2}−D_{2} have canonical singularities. Assume that π : X_{1} → X_{2} is a finite dominant morphism so that

π^{∗}(K_{X_{2}} + D_{2}) = K_{X_{1}} + D_{1}.

Let L be a big divisor on X_{2}. Then a(L, D_{2}) = a(π^{∗}(L), D_{1}).

In fact, it suffices to assume that either (X_{1}, D_{1}) or (X_{2}, D_{2}) satisfies the singularity condition [17, Sect. 20.3]. Proof. Given a finite dominant morphism π : X_{1} → X_{2} and a ℚ-Cartier divisor M on X_{2}, M is effective iff π^{∗}(M) is effective. Indeed, the divisor π_{∗}π^{∗}M is defined and equal to deg(π)M. Combining this with Proposition 1.3 gives the result. □

Remark 1.5 Let (X, D) be a log terminal pair so that X − D has singularities which are not canonical. Then our definition of the Kodaira energy differs slightly from Fujita’s [10]. In applications to integral points, we are interested in invariants of the open variety X − D. In Fujita’s definition, on passing from (X, D) to a good resolution, any exceptional divisors over X−D with negative discrepancy must be added to the boundary. This changes the open variety.

1.2 Integral points

Retain the notation from the previous section and assume that X and D are defined over a number field F. Let 𝔬_{S} denote the ring of integers of F, where S is a finite set of nonarchimedean places of F. Fix models X and D flat and proper over the ring of integers 𝔬_{S}. A (D, S)-integral point is an 𝔬_{S}-point of (X − D). In particular, if D = ∅ an integral point is the same as a rational point on X.

Let L be a very ample metrized line bundle on X, U ⊂ X a Zariski open subset and U a model of U over 𝔬_{S}. Let S be a finite set of places in F, including the archimedean places. Let

N(U, L, B) := #{x ∈ U(𝔬_{S}) | H_{L}(x) ≤ B}

denote the number of (D, S) integral points on U of L-height bounded by B. A natural extrapolation of Vojta’s conjecture about integral and rational points on varieties of (log-)general type [27] and Batyrev-Manin conjectures about rational points of bounded height on Fano varieties [9, 1] would be:

Conjecture 1.6 For any ϵ > 0, there exists a dense Zariski open subset U ⊂ X such that

N(U, L, B) ≪ B^{a(L,D)+ϵ}

as B → ∞. If −(K_{X} + D) is big then

N(U, L, B) ≫ B^{a(L,D)−ϵ},

as B → ∞, at least after a suitable finite extension of F and S.

The statement is independent of the choice of S and the choice of a metrization on L.

Many precise results about asymptotics of rational and integral points are currently available (see, for example, [9, 2, 4, 21, 22, 6, 7, 8] and the references therein). As far as we know, Conjecture 1.6 is compatible with all of them. However, to actually check this compatibility one has to compute the geometric invariants of (some resolution of) the pair (X, D). In particular, one has to determine the effective cone. This can be a formidable task even for rational varieties, e.g., like the moduli space of pointed rational curves \overline{M}_{0,n} (see [14]).

1.3 Computing effective cones

Let X be a nonsingular projective variety, perhaps with an action by a finite group G. We review strategies for computing the G-invariant effective cone Λ_{eff}(X)^{G} and thus the effective cone of the quotient X/G (cf. [16]).

A curve class [C] ∈ N_{1}(X) is said to be nef if [C].D ≥ 0 for each D ∈ Λ_{eff}(X). A family of curves passing through the generic point of X is automatically nef. Indeed, consider a family C → B of integral projective curves in X and an irreducible codimension-one subvariety D ⊂ X. If, for generic b ∈ B, the fiber C_{b} ⊄ D, we have [C_{b}].D ≥ 0.

Fix a collection of effective divisors

Γ = {A_{1}, …, A_{m}}

which we expect to generate Λ_{eff}(X)^{G}. To prove that Γ generates the (Ginvariant) effective cone, it suffices to find a collection of nef (G-invariant) curve classes

Ξ = {C_{1}, …, C_{ℓ}}

so that the cone generated by Γ contains the dual to the cone generated by Ξ.

In section 4, we shall use a refinement of this method (see [5], [24]). A divisor D ∈ Λ_{eff}(X) is moving relative to Γ if some multiple of D contains no element of Γ as a fixed component. Every effective divisor is a sum

M + ∑_{i=1}^{m} A_{i}d_{i}, d_{i} ≥ 0

where M is moving relative to Γ. To prove that Γ generates the effective cone, it suffices to show that M is an effective sum of the A_{i}.

A curve class is nef relative to Γ if [C].M ≥ 0 for each M which is moving relative to Γ. Any family of curves passing through the generic point of some A_{i} is nef relative to Γ. Consequently, to show that Γ generates the effective cone, it suffices to find a collection Ξ of curve classes, nef relative to Γ, so that the cone generated by Γ contains the dual to the cone generated by Ξ.

2 Construction of resolutions

2.1 Binary forms and SL_{2}-orbit closures

Let V be a two-dimensional vector space with coordinates z and w, equipped with the standard SL_{2}-action. Let Sym^{n}V^{∗} be the space of binary forms of degree n

f = x_{0}z^{n} + x_{1}z^{n−1}w + … + x_{n}w^{n}.

It carries an induced action of SL_{2} by substitution.

Associating to each form f ≠ 0 its roots α_{1}, …, α_{n} yields a map

(Sym^{n}V^{∗} − 0) → ℙ(V)^{n}/𝔖_{n}

and an identification ℙ(Sym^{n}V^{∗}) ≃ ℙ(V)^{n}/𝔖_{n}. The discriminant of a polynomial f is a homogeneous form in its coefficients x_{0}, …, x_{n} and defines a divisor D ⊂ X = ℙ(Sym^{n}V^{∗}).

Now we may state our main result:

Theorem 2.1 (Computation of Kodaira Energy) Let f be a generic bilinear form of degree n, X_{f} ⊂ ℙ(Sym^{n}V^{∗}) the closure of the SL_{2}-orbit through f, D_{f} the intersection of the discriminant with X_{f}, and L the restriction of the standard polarization to X_{f}. Then we have

a(L, D_{f}) = 2/n.

In particular, Conjecture 1.6 is consistent with Theorem 0.1.

To prove this, we require a resolution (i.e., a partial desingularization) of (X_{f}, D_{f}) on which we may evaluate a(L, D_{f}) using Proposition 1.3. This resolution will be induced by a natural resolution of (X, D).

Remark 2.2 Example 1.2 shows that (X, D) is far from being log-canonical. When n = 3, the discriminant has cusps in codimension one: a transverse slice

z^{3} + bzw^{2} + cw^{3}

intersects the discriminant in the cuspidal curve

4b^{3} + 27c^{2} = 0.

Our resolution of (X, D) will be a 𝔖_{n}-quotient of a natural desingularization for (ℙ(V)^{n}, Δ), where Δ is the diagonal, i.e., the points lying over the discriminant. Both admit interpretations as moduli spaces of stable maps.

2.2 Moduli spaces

Fix an integer n ≥ 3. Let \overline{M}_{0,n} denote the Knudsen-Mumford moduli space of stable curves of genus zero with n marked points [18]

(C, p_{1}, …, p_{n}).

Let \overline{M}_{0,n}(ℙ^{1}, 1) denote the Kontsevich moduli space of stable maps of degree one from genus-zero curves with n marked points to ℙ^{1} [19, 20, 12]

(C, p_{1}, …, p_{n}, μ : C → ℙ^{1}).

This is naturally isomorphic to the Fulton-MacPherson [11] configuration space ℙ^{1}[n] for n points in ℙ^{1} (see [12] §0). However, for our purposes it is convenient to use the moduli space notation.

We have the following natural maps:

1. the evaluation map

\overline{M}_{0,n}(ℙ^{1}, 1) ⟶ (ℙ^{1})^{n};
(C, p_{1}, …, p_{n}, μ) ↦ (μ(p_{1}), … μ(p_{n}));

2. forgetting the point p_{j}

ϕ_{j} : \overline{M}_{0,n} ⟶ \overline{M}_{0,n−1}, (n ≥ 4)
(C, p_{1}, …, p_{n}) ↦ (C′, p_{1}, …, p̂_{j}, …, p_{n});
ϕ_{j} : \overline{M}_{0,n}(ℙ^{1}, 1) ⟶ \overline{M}_{0,n−1}(ℙ^{1}, 1)
(C, p_{1}, …, p_{n}, μ) ↦ (C′, p_{1}, …, p̂_{j}, …, p_{n}, μ′).

3. taking projective equivalence classes

ψ : \overline{M}_{0,n}(ℙ^{1}, 1) ⟶ \overline{M}_{0,n}
(C, p_{1}, …, p_{n}, μ) ↦ (C′, p_{1}, …, p_{n}).

C′ is obtained from C by ‘collapsing’ the irreducible components which are destabilized when p_{j} (resp., the polarization) is removed.

Finally, we enumerate the boundary divisors of these moduli spaces. For each partition

{1, …, n} = S ∪ S′, 2 ≤ |S| ≤ |S′| ≤ n − 2,

consider stable curves

C = (ℙ^{1}, p_{j}, j ∈ S) ∪ (ℙ^{1}, p_{j}, j ∈ S′),

which form a divisor δ_{S,S′} ⊂ \overline{M}_{0,n}. The union of these is denoted δ. Note that the 𝔖_{n}-orbits of {δ_{S,S′}} correspond to the integers

|S| = 2, …, ⌊n/2⌋.

For each subset

S ⊂ {1, …, n}, 2 ≤ S

consider stable maps

μ : C = (ℙ^{1}, p_{j}, j ∈ S) ∪ (ℙ^{1}, p_{j}, j ∈ S′) ⟶ ℙ^{1}

collapsing the first component and mapping the second isomorphically onto ℙ^{1}. These form a divisor B_{S} ⊂ \overline{M}_{0,n}(ℙ^{1}, 1). The 𝔖_{n}-orbits of \{B_{S}\} correspond to integers

s = |S| = 2, …, n

and we define

B[s] := ∑_{|S|=s} B_{S} and B := ∑_{s=2}^{n} B[s]

Theorem 2.3 The moduli spaces \overline{M}_{0,n}(ℙ^{1}, 1) and \overline{M}_{0,n} are smooth projective algebraic varieties. Moreover, the boundary is a divisor with strict normal crossings.

Remark 2.4 In particular, the pair (\overline{M}_{0,n}(ℙ^{1}, 1), B) is log-canonical.

2.3 Resolution for the full moduli space

We obtain a good resolution (X̃, D̃) of (X, D) using the above formalism. Consider the quotient map

q : \overline{M}_{0,n}(ℙ^{1}, 1) → X̃ := \overline{M}_{0,n}(ℙ^{1}, 1)/𝔖_{n}.

Let D̃[s] and D̃ be the images of B[s] and B under this map.

Proposition 2.5 The map q is ramified only along the boundary B. At the generic points of B[2] the ramification has order 2. For all s = 3, .., n, the map q is unramified at the generic points of B[s]. We have the formula

q^{∗}(KX̃ + D̃) = K_{\overline{M}_{0,n}(ℙ^{1},1)} + B

and (X̃, D̃) is log-canonical.

Proof. The map q ramifies at points corresponding to stable maps

(C, p_{1}, ..., p_{n}, μ)

that admit an automorphism permuting the marked points. The ramification order is the order of this automorphism group. If the images of the n points under μ are distinct then there is no automorphism of μ permuting them. This proves the first assertion. If marked points coincide there is an irreducible component ℙ^{1} ⊂ C which is collapsed by μ and which contains these points. If there are two such points this component admits an automorphism of order two exchanging the points and fixing the point of intersection with the rest of C. This proves the second assertion. If there are more than two marked points then there is generally no such automorphism. This proves the third assertion.

The ramification formula and the fact that the pair (X̃, D̃) is log-canonical follow from an easy local computation combined with Remark 2.4 (see Propositions 20.2 and 20.3 of [17]). □

Take 𝔖_{n}-quotients of the point map to obtain a birational map

ϱ : X̃ → ℙ(Sym^{n}V^{∗}),

assigning to p_{1}, ..., p_{n} ∈ ℙ^{1} a polynomial vanishing at these points. The boundary divisor D̃[2] is the proper transform of the discriminant D under ϱ. The boundary divisors D̃[s] (for s ≥ 3) are the exceptional divisors for ϱ.

2.4 Resolution of the generic orbit

Let α := (α_{1}, ..., α_{n}) be a set of distinct complex numbers and f = f_{α} the binary form of degree n with roots α_{j}. Let C_{α} ∈ M_{0,n} be the corresponding pointed rational curve and μ_{α} ∈ M_{0,n}(ℙ^{1}) the corresponding map. The fiber

Y_{α} := ψ^{−1}(C_{α}) ⊂ \overline{M}_{0,n}(ℙ^{1}, 1)

contains μ_{α}. Let X̃_{f} be the image of Y_{α} under the quotient map q and D̃_{f} its intersection with the boundary D̃. This coincides with the general fiber of the map

ψ′ : X̃ = \overline{M}_{0,n}(ℙ^{1}, 1) → \overline{M}_{0,n}/𝔖_{n}.

The map ϱ induces a resolution

ϱ_{f} : X̃_{f} → X_{f},

with ϱ_{f}(D̃_{f}) = D_{f}.

To describe the Y_{α} explicitly, we use the tower

[figure: \overline{M}_{0,n}(ℙ^{1}, 1); ϕ_{n}; ψ; _{\overline{M}_{0,n}}; ϕ_{n}; \overline{M}_{0,n−1}(ℙ^{1}, 1); ψ; _{\overline{M}_{0,n−1}}; ϕ_{4}; ϕ_{4}; \overline{M}_{0,3}(ℙ^{1}, 1); ψ; \overline{M}_{0,3}.]

When n = 3, \overline{M}_{0,3} = point and Y_{α} ≃ \overline{M}_{0,3}(ℙ^{1}, 1), which is isomorphic to the product (ℙ^{1})^{3} blown up along the small diagonal Δ_{small}. The boundary divisors correspond to the following stable maps

B[2] =  , B[3] =  .

In the above pictures the collapsed components are represented by vertical lines. Note that the normal bundle

N_{Δ} ≅ O(2) ⊕ O(2),

so that the exceptional divisor E = B[3] ≃ ℙ^{1} × ℙ^{1}. Let

π_{1} : E → ℙ^{1}

be the projection to the cross ratio of the marked points and the node and

π_{2} : E → ℙ^{1}

the projection onto the image of the collapsed curve.

The divisor B[2] is the proper transform of Δ, the large diagonal.

For the arbitrary degree case, we analyze the failure of the block squares in the tower to be fiber products. Given a generic

C_{α} = (ℙ^{1}, α_{1}, …, α_{n}) ∈ M_{0,n}, α_{i} ≠ α_{j},

we compare the fibers Y_{α_{1},...,α_{n}} = ψ^{−1}(C_{α}) and Y_{α_{1},...,α_{n−1}} = ψ^{−1}(ϕ_{n}(C_{α})) = ψ^{−1}(ℙ^{1}, α_{1}, …, α_{n−1}) using the forgetting map

ϕ_{n} : \overline{M}_{0,n}(ℙ^{1}, 1) → \overline{M}_{0,n−1}(ℙ^{1}, 1).

Given a stable map

(C, α_{1}, …, α_{n}, μ) ∈ ψ^{−1}(C_{α}),

there are three cases to consider:

1. C = ℙ^{1};

2. C = ℙ^{1} ∪ ℙ^{1} with the collapsed component containing α_{1}, ..., α_{n};

3. C = ℙ^{1} ∪ ℙ^{1} with the collapsed component containing α_{1}, ..., α_{n−1} but not α_{n}.

Case 1:  ⟶^{ψ}

Case 2:  ⟶^{ψ}

Case 3:  ⟶^{ψ}

Over the open subset of Y_{α_{1},...,α_{n}} corresponding to the first two cases, ϕ_{n} induces an isomorphism between Y_{α_{1},...,α_{n}} and Y_{α_{1},...,α_{n−1}}. In the third case, we forget the image of the n-th marked point. The map ϕ_{n} blows up the locus in Y_{α_{1},...,α_{n−1}} where α_{1}, ..., α_{n−1} are on the collapsed component and α_{n} coincides with the node (of attachment). This is a curve isomorphic to ℙ^{1} ⊂ (B[n − 1] ∩ Y_{α_{1},...,α_{n−1}}): The generic map takes the form:

We summarize the above discussion in the following

Proposition 2.6 Let α_{1}, ...α_{n} be distinct complex numbers. The forgetting maps induce a sequence of birational morphisms

Y_{α_{1},...,α_{n}} ⟶^{ϕ_{n}} Y_{α_{1},...,α_{n−1}} … ⟶^{ϕ_{4}} Y_{α_{1},α_{2},α_{3}} ≃ \overline{M}_{0,3}(ℙ^{1}, 1).

The moduli space of stable maps \overline{M}_{0,3}(ℙ^{1}, 1) is isomorphic to (ℙ^{1})^{3} blown up along the small diagonal with exceptional divisor E ≃ ℙ^{1} × ℙ^{1}. The map ϕ_{j} blows up the proper transform of π_{1}^{−1}(α_{j}). In particular, Y_{α_{1},...,α_{n}} is smooth and its boundary has strict normal crossings, contained in B[n − 1] ∪ B[n].

Remark 2.7 We are blowing up along disjoint curves, so the order of the blow-up does not matter.

Proposition 2.8 Let f be a generic binary form of degree n ≥ 3 with roots α_{1}, ..., α_{n}. Then the restriction of q to X_{f} is ramified only along the boundary B ∩ X_{f}. At generic points of (B[n] ∪ B[n − 1]) ∩ Y_{α}, the restriction of q is unramified. We have the formula

q^{∗}(KX̃_{f} + D̃_{f}) = K_{Y_{α}} + [Y_{α} ∩ B]

and (X̃_{f}, D̃_{f}) is log-canonical.

Proof. The argument is similar to the one in Proposition 2.5, and is omitted. □

3 Verification of exponents

3.1 Explicit basis of Pic(Y_{α})

Write

Pic((ℙ^{1})^{3}) = ℤg_{1} + ℤg_{2} + ℤg_{3}, g_{i} = pr_{i}^{∗}(c_{1}(O_{ℙ^{1}}(+1))),

with large diagonals

Δ_{ij} = g_{i} + g_{j} − E, B[2] = 2(g_{1} + g_{2} + g_{3}) − 3E.

By Proposition 2.6, Y_{α} is obtained by blowing up the (n − 3) sections of

π_{2} : B[3] → ℙ^{1}.

Let F_{4}, ..., F_{n} denote the corresponding exceptional divisors and identify E and its proper transform. Relabel

\begin{aligned}F_{k} & = & Δ_{ij}, \{i, j, k\} = [1, 2, 3] \\ & = & g_{i} + g_{j} − E − F_{4} − ⋯ − F_{n}, \end{aligned}

so that 𝔖_{n} acts on the F_{k}, k = 1, ..., n, in the obvious way. Note that E and the F_{k} generate Pic(Y_{α}).

Proposition 3.1 The 𝔖_{n}-stable boundary divisors

\begin{aligned}A[n − 1] & = & F_{1} + ⋯ + F_{n} \\ A[n] & = & E, \end{aligned}

generate the 𝔖_{n}-invariant Picard group of Y_{α}, and A[j] = B[j] ∩ Y_{α}. The canonical class of Y_{α} is

\begin{aligned}K & = & −2(g_{1} + g_{2} + g_{3}) + E + 2(F_{4} + ⋯ + F_{n}) \\ & = & −A[n − 1] − 2A[n] \end{aligned}

3.2 Computation of the effective cone

Lemma 3.2 The 𝔖_{n}-invariant effective cone of Y_{α} is generated by the classes A[n] and A[n − 1].

Proof. We apply the method of §1.3. The class A[n] is exceptional and thus a generator of the effective cone. To show that A[n−1] is the second generator, we exhibit a nef curve not intersecting A[n − 1]. Consider the 𝔾_{m}-action on ℙ^{1}:

ρ_{t} : (z, w) ↦ (tz, w).

We may assume that the points α_{1}, ..., α_{n} are not contained in the fixed point locus of ρ_{t}. Any singular element in the orbit closure is:

where the point of attachment is 0 (or ∞) and the other labelled point is ∞ (resp. 0). This is disjoint from A[n − 1]. □

3.3 Proof of Theorem 2.1

Proof. The Kodaira energy for (X_{f}, D_{f}) can be computed on (X̃_{f}, D̃_{f}), by Propositions 1.3 and 2.8. By Propositions 1.4 and 2.6, it suffices to compute the Kodaira energy for (Y_{α}, A[n] + A[n − 1]). Recall there is a composed morphism

β : Y_{α} →^{q} X_{f} ↪ ℙ(Sym^{n}V^{∗}) ≃ ℙ^{n}.

Lemma 3.3 The pull-back of the hyperplane class takes the form

L =: [β^{∗}O_{ℙ^{n}}(+1)] = \frac{1}{2} ((n − 2)A[n − 1] + nA[n]) .

Proof. Let R be the class of a curve in A[n] corresponding to

with varying point of attachment on the collapsed component. This is the proper transform of the generic fiber of the map π_{1} : E → ℙ^{1}. Then

\begin{aligned}A[n − 1]|_{R} & = & n = # \{ , j = 1, …, n\} \\ A[n]|_{R} & = & −1 − (n − 3) = 2 − n. \end{aligned}

For the second intersection number, note that A[n] = E and apply the blowup description of Proposition 2.6. In \overline{M}_{0,3}(ℙ^{1}, 1) we have

E = ℙ(N_{Δ_{small}}) = ℙ^{1} × ℙ^{1}

and N_{E} = O(−1). After blowing up (n − 3) further sections of

E → Δ_{small}

the normal bundle is reduced to O(−1 − (n − 3)).

We know that A[n] = B[n] ∩ Y_{α} is collapsed by the map β, so

β^{∗}O(1) = c ((n − 2)A[n − 1] + nA[n])

for some c ∈ ℕ. Since

(n − 2)A[n − 1] + nA[n] = 2 ((n − 2)(g_{1} + g_{2} + g_{3}) − (n − 3)E − (n − 2)(F_{4} + ⋯ + F_{n})) .

the claim follows. □

We have

K_{Y_{α}} + [Y_{α} ∩ B] = K_{Y_{α}} + A[n] + A[n − 1] = −A[n]
K_{Y_{α}} + [Y_{α} ∩ B] + αL = α\frac{n − 2}{2}A[n − 1] + (α\frac{n}{2} − 1)A[n],

and by definition

a(β^{∗}L) := inf{α | αβ^{∗}L + K_{Y_{α}} + [Y_{α} ∩ B] ∈ Λ_{eff}(Y_{α})}.

Hence Lemma 3.2 yields

a(L, [Y_{α} ∩ B]) = 2/n.

Thus a(L, D_{f}) = 2/n, as desired! □

4 The 𝔖_{n}-invariant effective cone of the full moduli space

In this section, we compute the 𝔖_{n}-invariant part of the effective cone of \overline{M}_{0,n}(ℙ^{1}, 1), its canonical class, and the Kodaira energy of the line bundle L := β^{∗}O_{ℙ^{n}}(+1), where

β = ρ ∘ q : \overline{M}_{0,n}(ℙ^{1}, 1) →^{q} X̃ →^{ρ} ℙ^{n}.

We will also compute the Kodaira energy of H := ρ^{∗}O_{ℙ^{n}}(+1).

We first recall some basic facts about \overline{M}_{0,n}(ℙ^{1}, 1) ≃ ℙ^{1}[n], following [11]. In addition to the divisor classes B_{S} introduced above, we shall also consider

L_{a} := \{(C, p_{1}, …, p_{n}, μ) ∈ \overline{M}_{0,n}(ℙ^{1}, 1) : μ(p_{a}) = 0 ∈ ℙ^{1}\}, a = 1, …, n.

The cohomology H^{∗}(\overline{M}_{0,n}(ℙ^{1}, 1)) is generated by the classes L_{a} and B_{S} , subject to the relations

1. L_{a}^{2} = 0;

2. B_{S} · B_{S′} = 0 for S ∩ S′ ≠ ∅;

3. (L_{a} − L_{a′})B_{S} = 0 for a, a′ ∈ S;

4. (∑_{S⊃{a,a′}} B_{S}) = L_{a} + L_{a′}, for 1 ≤ a < a′ ≤ n.

The generators of the 𝔖_{n}-invariant subspace are

L := ∑_{a=1}^{n} L_{a}, B[s] = ∑_{|S|=s} B_{S}, 2 ≤ s ≤ n.

After averaging over 𝔖_{n}

(n − 1)L = ∑_{s=2}^{n} \frac{s(s − 1)}{2}B[s].  (1)

Theorem 4.1 The classes D[2], ..., D[n] generate the effective cone of Y. The classes B[2], ..., B[n] generate the 𝔖_{n}-invariant effective cone of the moduli space \overline{M}_{0,n}(ℙ^{1}, 1).

Proof. We implement the strategy of §1.3 with

Γ = {B[2], …, B[n]}.

This entails finding curve classes that are nef relative to Γ. Let

M = ∑_{j=2}^{n} d_{j}B[j]

denote an 𝔖_{n}-invariant divisor class with no boundary divisors as fixed components.

Recall the description of the boundary divisor B_{S}:

\begin{aligned}B_{S} & ≃ & \overline{M}_{0,n+1−s}(ℙ^{1}, 1) × \overline{M}_{0,s+1}, & s = |S| > 2, \\ & ≃ & \overline{M}_{0,n−1}(ℙ^{1}, 1), & s = 2. \end{aligned}

Take s ≥ 3 and let C_{s} ⊂ B_{S} be the class of the generic fiber of the map

\overline{M}_{0,n+1−s}(ℙ^{1}, 1) × \overline{M}_{0,s+1} → \overline{M}_{0,n+1−s}(ℙ^{1}, 1) × \overline{M}_{0,s}

forgetting the attaching point. Since C_{s} passes through the generic point of B_{S}, averaging C_{s} over 𝔖_{n} yields a curve class which is nef relative to Γ. In particular, for each 𝔖_{n}-invariant divisor M = ∑_{j=2}^{n} d_{j}B[j], moving relative to Γ, we have C_{s} · M ≥ 0.

We compute intersections of C_{s} with the various elements of Γ. First, the map β blows down the divisors B_{S} for |S| ≠ 2; the data of the collapsed component is lost completely. It follows that L · C_{s} = 0. A simple combinatorial analysis gives

C_{s} · B_{T} = \{ \begin{matrix}1 & if T = S − \{σ\}; \\ 0 & otherwise, unless T = S. \end{matrix}

which means that C_{s} · B[s − 1] = s. Relation 1 gives

0 = B_{S} · C_{s} \frac{s(s − 1)}{2} + s\frac{(s − 1)(s − 2)}{2}

so B_{S} · C_{s} = −(s − 2). To summarize, we have

C_{s} · B[j] = \{ \begin{matrix}s & if j = s − 1; \\ −(s − 2) & if j = s; \\ 0 & otherwise. \end{matrix}

Using this information, we extract inequalities on the coefficients of M. The condition M · C_{s} ≥ 0 yields

sd_{s−1} ≥ (s − 2)d_{s},

so we get a chain of inequalities:

d_{n} ≤ \frac{n}{n − 2}d_{n−1} ≤ \frac{n(n − 1)}{(n − 2)(n − 3)}d_{n−2} ≤ … ≤ \frac{n(n − 1)}{2}d_{2}.  (2)

If some d_{s} < 0 then d_{j} < 0 for each j ≥ s.

We consider another curve class in B_{S} to get inequalities in the reverse direction. Fix s ≥ 2 and let R_{s} denote the class of the generic fiber of

\overline{M}_{0,n+1−s}(ℙ^{1}, 1) × \overline{M}_{0,s+1} → \overline{M}_{0,n−s}(ℙ^{1}, 1) × \overline{M}_{0,s+1}

induced by forgetting τ, one of the n+1−s points not contained in S. Again, R_{s} passes through the generic point of B_{S}, so averaging over 𝔖_{n} yields a curve class such that R_{s} · M ≥ 0.

We compute intersections as before. The map β sends R_{s} to a line in ℙ^{n}, i.e., the linear forms with n − 1 fixed roots and one varying root. It follows that L · R_{s} = 1. The line R_{s} intersects B_{T} properly in the following cases

R_{s} · B_{T} = \{ \begin{matrix}1 & if T = S ∪ \{τ\}; \\ 1 & if T = \{τ, υ\}, υ ∉ S; \\ 0 & otherwise, unless T = S. \end{matrix}

Summing over 𝔖_{n}-orbits gives

R_{s} · B[j] = \{ \begin{matrix}1 & if j = s + 1; \\ n − s − 1 & if j = 2; \\ 0 & otherwise, unless j = s. \end{matrix}

Applying Relation 1, we find

(n − 1) = s(s − 1)/2R_{s} · B[s] + (s + 1)s/2 + (n − s − 1),

so R_{S} · B_{S} = R_{s} · B[s] = −1.

We extract the inequalities

d_{s+1} − d_{s} ≥ (n − s − 1)d_{2};

in particular, if d_{2} > 0 then each d_{j} > 0. Adding together the inequalities

d_{n} − d_{n−1} ≥ 0
d_{n−1} − d_{n−2} ≥ d_{2}
…
d_{4} − d_{3} ≥ (n − 4)d_{2}
d_{3} ≥ (n − 2)d_{2}.

gives

d_{n} ≥ \frac{n^{2} − 5n + 8}{2}d_{2}.

Combining with inequality 2, we obtain

(n^{2} − 1)d_{2} ≥ (n^{2} − 5n + 8)d_{2},

hence d_{2} > 0. □

Theorem 4.2 On the moduli space of stable maps (\overline{M}_{0,n}(ℙ^{1}, 1), B) and its 𝔖_{n}-quotient (Y, D), we have

a(B, L) = a(D, H) = 2/n.

Proof. We proceed to calculate the canonical class:

K_{\overline{M}_{0,n}(ℙ^{1},1)} = −2L + ∑_{s=3}^{n} B[s](s − 2)

This follows from the explicit blowup realization of the Fulton-MacPherson configuration space ℙ^{1}[n] = \overline{M}_{0,n}(ℙ^{1}, 1) [11]. The exceptional divisors B_{S} (for s = |S| ≥ 3) arise from blowing up centers in codimension s − 1.

We compute the log-Kodaira energy of L on \overline{M}_{0,n}(ℙ^{1}, 1) with respect to the boundary B. We have

K + B + aL = (a − 2)L + ∑_{s=2}^{n}(s − 1)B[s]

= (a − 2) (∑_{s=2}^{n} \frac{s(s − 1)}{2(n − 1)}B[s]) + ∑_{s=2}^{n}(s − 1)B[s]

= ∑_{s=2}^{n}(s − 1) (\frac{s(a − 2)}{2(n − 1)} + 1) B[s]

which is effective if and only if

a ≥ \frac{2(s − n + 1)}{s}, s = 2, …, n.

The most restrictive inequality occurs when s = n, where we obtain a ≥ 2/n. Consequently, a(L, B) = 2/n, as expected.

The Kodaira energy for (X, D) can be computed on (X̃, D̃), by Propositions 1.3 and 2.5. By Proposition 1.4 and Theorem 2.3, it is equal to the Kodaira energy for (\overline{M}_{0,n}(ℙ^{1}, 1), B). □

5 Final remarks

A) The orbit closure X_{f} depends on the form [f]. We get equivariant compactifications of PGL_{2} depending on moduli. This dependence is made abundantly clear in the blow-up description of Proposition 2.6.

B) The pair (\overline{M}_{0,n}, δ) is of log general type: K_{\overline{M}_{0,n}} + δ is ample and log canonical (see, for example, §7.1 of [13]). The map

ψ : \overline{M}_{0,n}(ℙ^{1}, 1) → \overline{M}_{0,n}

is a log-Fano fibration onto a log-variety of general type (or the point, when n = 3).

C) The pair (\overline{M}_{0,n}, δ) satisfies Vojta’s conjecture. We realize M_{0,n} as an open subset of an algebraic torus with explicit complement. Fix n − 1 points in ℙ^{n−3} in general position. Consider the set H of \frac{1}{2}(n−1)(n−2) hyperplanes spanned by n − 3 of the fixed points. Kapranov [15] has shown that M_{0,n} ≃ ℙ^{n−3} − ∪_{H∈H}H. The torus is obtained by excising the n − 2 hyperplanes spanned by subsets of the first n − 2 of the points.

D) We therefore expect that the asymptotic behavior of integral points of \overline{M}_{0,n}(ℙ^{1}, 1) with respect to the boundary is obtained by summing the contributions of the integral points on the fibers of ψ. This explains why the Kodaira energies for the moduli space of stable maps (Theorem 4.2) and the fibers of ψ (Theorem 2.1) should coincide (for the case of rational points, see [2]).

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Brendan Hassett Department of Mathematics Rice University, MS 136 6100 S. Main Street Houston, TX 77005-1892, U.S.A. hassett@math.rice.edu

Yuri Tschinkel Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NJ 08544-1000, U.S.A. ytschink@math.princeton.edu