Local stable reduction of plane curve singularities

Brendan Hassett ^{∗}

December 21, 1999

Abstract

Consider a family of curves over the disc, with smooth fibers except for the central fiber over the origin. By the local stable reduction theorem, after suitable blow-ups and base changes we obtain a family such that the central fiber has reduced normal crossings. This stable central fiber has two parts: the proper transform of the original central fiber and the ‘tail’. Which tails arise when the original central fiber is a given plane curve singularity? We address this question using the technique of stable reduction for log surfaces. For certain singularities, we find that weighted plane curves naturally arise as tails.

1991 AMS Subject Classification: Primary 14H20, Secondary 14J10

1 Introduction

One important consequence of the log minimal model program is the existence of compactifications for moduli spaces of log surfaces of general type. These compactifications are discussed in [16] and [1], and may be regarded as higher-dimensional analogs of moduli spaces of pointed stable curves. Implicit in these compactifications is a notion of stable reduction, i.e. a procedure for modifying a family of log surfaces acquiring arbitrarily complicated singularities to obtain a family of log surfaces with certain prescribed singularities. For instance, after stable reduction the boundary of a log surface has only nodal singularities. Unfortunately, this process is not well-understood in practice, even in the simplest cases.

∗The author was partially supported by a National Science Foundation Postdoctoral Fellowship.

In this paper, we consider stable reduction from a local standpoint. Given a germ of a family of log surfaces degenerating to a singular pair (S_{0}, C_{0}), what kinds of surfaces appear in the course of stable reduction? The simplest cases to consider are germs of isolated plane curve singularities. One attractive aspect of these special cases is that they give insight into the following elementary question: which stable curves are the limits of plane curves? Another, more ‘modern’ formulation of this question is: which stable maps to the plane are smoothable?

This paper is organized as follows. First, we review some basic properties of isolated plane curve singularities, their resolutions, and their classification. In particular, we introduce singularities of toric type (e.g. x^{p} − y^{q} = 0) and quasitoric type (e.g. xy(x^{p} − y^{q}) = 0); these include all the simple ADE singularities. In section three, we describe the notion of local stable reduction for curves and log surfaces, and prove some elementary properties of the stable limits. In section four, we give a partial description of these stable limits, the Main Component Theorem (Theorem 4.1). This is made explicit for singularities of toric and quasitoric type. In the fifth section, we describe some degenerations of surfaces that arise naturally from the log minimal model program. In the sixth section, we use these degenerations to describe the ‘tails’ of stable limits arising from smoothings of singularities of toric and quasitoric type. The key statements are Theorems 6.2 and 6.3, Proposition 6.4, and Conjectures 6.5 and 6.6. Certain examples, like the simple ADE singularities, are discussed in more detail. In the last section, we show how these ideas shed light on the geometry of the equisingular deformation space. We also describe certain boundary components of the compatification for the moduli space of pairs (ℙ^{2}, C), where C is a smooth plane curve of degree d ≥ 4. In a subsequent paper [13], we enumerate the stable limits of pairs (ℙ^{2}, C), where C is a smooth plane quartic.

I would like to thank Andrew Kresch for some helpful conversations on toric geometry and Rahul Pandharipande for insights on weighted projective spaces. Dan Edidin and Joe Harris generously explained their ideas on stable reduction for double points. Francisco Gallego and B. P. Purnaprajna graciously shared their examples of stable limits of log surfaces. I have also benefitted from discussions with Lawrence Ein and Anatoly Libgober, and from William Fulton’s lucid lectures on toric varieties. Henry Pinkham made helpful comments on the manuscript and the connection between Theorem 6.2 and his work [21]. Alessio Corti has informed me that he and his students independently discovered some of the results discussed here.

Throughout this paper, we work over ℂ.

2 Singularities of toric and quasitoric type

Throughout this paper, C_{0} ⊂ S_{0} = Spec ℂ[[x, y]] denotes the germ of an isolated reduced plane curve singularity at the origin x = y = 0.

Definition 2.1 An imbedded resolution of C_{0} is a projective birational morphism of smooth surfaces r : P_{1} → S_{0}, with exceptional locus E, such that:

1. C_{1}, the proper transform of C_{0}, is smooth;

2. E ∪ C_{1} is a normal crossings divisor.

An imbedded resolution (P_{1}, C_{1}) is minimal if every other imbedded resolution factors through it.

Since r is a birational projective morphism of smooth surfaces, it may be obtained by a succession of blow-ups of smooth points. The exceptional locus E is thus a tree of smooth rational curves {E_{i}}. The intersection form on P_{1} restricts to a negative definite unimodular quadratic form on ⊕_{i}ℤE_{i}.

Proposition 2.2 Let C_{0} ⊂ Spec ℂ[[x, y]] be a plane curve singularity. Its minimal imbedded resolution (P_{1}, C_{1}) is characterized by the fact that each irreducible exceptional curve E_{i} ⊂ P_{1} satisfies at least one of the following:

1. E_{i}^{2} < −1;

2. (C_{1} + E − E_{i})E_{i} > 2.

Given an arbitrary imbedded resolution, the minimal resolution is obtained by blowing down all exceptional curves satisfying neither of these properties. Evidently, these curves may be blown down without introducing any unwanted singularities.

Figure 1: Topological type of y^{4} = x^{10}

Let C_{0} be an isolated plane curve singularity and let (P_{1}, C_{1}) be its minimal imbedded resolution. Let E_{1}, E_{2}, …, E_{n} denote the distinct irreducible components of the exceptional locus. Consider the graph Γ_{C_{0}} with edges corresponding to the E_{i} and vertices corresponding to nonempty intersections E_{i} ∩ E_{j} with i ≠ j. Clearly Γ_{C_{0}} is a tree with two edges meeting at each vertex. The edges of Γ_{C_{0}} have two natural labellings

Σ_{C_{0}} : E_{i} → E_{i}^{2} Ξ_{C_{0}} : E_{i} → E_{i}C_{1}

by negative and non-negative integers respectively.

Definition 2.3 The topological type of C_{0} is defined as the labelled tree (Γ_{C_{0}}, Σ_{C_{0}}, Ξ_{C_{0}}). Two curves C_{0} and C′_{0} have the same topological type if there is a bijection Γ_{C_{0}} ≃ Γ_{C′_{0}} compatible with the labellings.

Of course, most triples (Γ, Σ, Ξ) are not realized as the topological type of any singularity.

We shall often use diagrams to represent the topological type of the singularities we consider. On these diagrams, Γ_{C_{0}} is represented as a tree of straight line segments, the values of Σ_{C_{0}} are indicated by the numbers in parentheses, and the values of Ξ_{C_{0}} by the number of intersections of the proper transform C_{1} with the edges of the tree. For example, the topological type of y^{4} = x^{10} is given in figure 1.

Intuitively, an equisingular deformation of an isolated plane curve singularity C_{0} is a deformation of C_{0} such that all the fibers have the same topological type as C_{0}. A rigorous definition was given by Wahl [22].

Our notion of topological type coincides with other concepts used to characterize the topology of the singularity, like Puiseux pairs and multiplicity sequences. See [4] chapter 8 for an introduction to these notions; [23] and [24] (Part I) show that all these definitions coincide.

Figure 2: Topological types of x(y^{4}−x^{10}) = 0, y(y^{4}−x^{10}) = 0, xy(y^{4}−x^{10}) = 0

Definition 2.4 Let p and q be integers with p ≥ q > 1. A plane curve singularity C_{0} is said to be of toric type (p, q) if it has the same topological type as x^{p} = y^{q}. We say that C_{0} is of quasitoric type if it has the same topological type as one of the following:

1. x(x^{p} − y^{q}) = 0 with p > q;

2. y(x^{p} − y^{q}) = 0 with p > q and q |̸p;

3. xy(x^{p} − y^{q}) = 0 with p > q and q |̸p;

The branches x = 0 and y = 0 are said to be distinguished.

The conditions on p and q guarantee that each of the equations above determines a distinct topological type. See figure 2 for diagrams representing certain singularities of quasitoric type. Remark: Note that the simple singularities of types A_{n}, D_{4}, E_{6}, and E_{8} are of toric type. Those of types D_{k}(k > 4) and E_{7} are of quasitoric type.

Proposition 2.5 If C_{0} is of type (p, q) then

1. Γ_{C_{0}} is a chain;

2. Ξ_{C_{0}} > 0 for exactly one component E_{n}, the unique component with Σ_{C_{0}} = −1.

Furthermore, E_{n} is an endpoint of the chain iff q|p.

Proposition 2.6 If C_{0} is of quasitoric type then

1. Γ_{C_{0}} is a chain; 2. the proper transforms of the non-distinguished branches all intersect a

single E_{n}, the unique component with Σ_{C_{0}} = −1;

3. the proper transforms of the distinguished branches meet components at the ends of the chain with Σ_{C_{0}} < −1 and Ξ_{C_{0}} = 1.

To prove the propositions, we recast them in inductive form:

Lemma 2.7 Assume p and q are integers with p ≥ q > 0. The minimal resolution of xy(x^{p} − y^{q}) = 0 satisfies the following:

1. The union of the exceptional curves and the proper transforms of the distinguished branches x = 0 and y = 0 forms a chain, with the distinguished branches at opposite ends of the chain.

2. The non-distinguished branches all meet a single exceptional component E_{n}, which is the exceptional curve of the last blow-up in the resolution and the unique component with Σ_{C_{0}} = −1.

3. The proper transform of x = 0 (resp. y = 0) intersects E_{n} iff q = p (resp. q|p).

After blowing up once we obtain the local equations

xt(x^{p−q} − t^{q}) = 0 sy(s^{p}y^{p−q} − 1) = 0

for the union of the exceptional divisors and the proper transform of C_{0}. Hence we proceed by induction on n, the number of blow-ups occurring in the minimal resolution. If n = 1 then p = q and the result is evident. For the induction, we simply reinterpret the local equations above. Consider the exceptional curve E_{1} and the proper transform of y = 0 as distinguished branches of xt(x^{p−q} − t^{q}) = 0. More precisely, set x_{1} = x and y_{1} = t if p − q ≥ q and x_{1} = t and y_{1} = x otherwise. The non-distinguished branches of this new singularity correspond with those of xy(x^{p} − y^{q}) = 0.

We extract the results on the minimal resolution. It is clear that only the last exceptional divisor E_{n} satisfies E_{n}^{2} = −1, and all the non-distinguished branches meet E_{n}. This yields (2). The proper transform of x = 0 (i.e. s = 0) intersects E_{1}, but none of the other exceptional components, and thus is at one end of the chain. By induction, the union of the proper transform of y = 0 (i.e. t = 0) and the exceptional locus also forms a chain, with E_{1} and the proper transform of y = 0 at opposite ends. Thus (1) follows. The proof of (3) is straightforward, and left to the reader. □

Proposition 2.8 Let C_{0} be a singularity with minimal resolution satisfying the conditions of Proposition 2.5 or 2.6. Then there exist integers p and q and a square-free weighted-homogeneous polynomial g_{0}(x, y) = x^{p}+…+y^{q} (where x and y have weights q and p respectively) so that C_{0} admits an equisingular specialization to one of the following:

1. g_{0}(x, y) = 0 with p ≥ q > 1;

2. xg_{0}(x, y) = 0 with p > q > 1;

3. yg_{0}(x, y) = 0 with p > q > 1 and q |̸p;

4. xyg_{0}(x, y) with p > q > 1 and q |̸p.

Again, the proposition is proved by induction on the number of exceptional curves in a minimal resolution. Number these curves E_{1}, …, E_{n} in the order they appear in the minimal resolution. If n = 1, then the singularity is evidently topologically equivalent to x^{p} = y^{p} for some p. Precisely, for some coordinates x and y transverse to all the branches of C_{0} = {f = 0}, the homogeneous part of f with minimal degree takes the form

g_{0}(x, y) = x^{p} + … + y^{p}.

Here g_{0} is square-free, and f = 0 has the same topological type as g_{0} = 0.

Now we establish the inductive step. Let C_{0} be a singularity with minimal resolution containing n > 1 exceptional divisors and satisfying the conditions of the Propositions 2.5 or 2.6. Blow up once, and consider C′ = C′_{0} ∪ E_{1}, the union of the proper transform and the exceptional curve. This has (at most) two distinct singularities r_{1} and r_{2}; r_{1} is where the non-distinguished branches meet E_{1} and r_{2} is where E_{1} meets a distinguished branch D. By induction, we know that the singularity of C′ at r_{1} specializes to one of the equations above, with E_{1} satisfying x′ = 0 or y′ = 0. Of course, C_{0} is obtained by contracting one of these lines, and the image of the branches through r_{1} satisfies equations 1 or 3 above. However, the image of the distinguished branch D (if it exists) satisfies some equation x + higher order terms = 0, so the equation of C_{0} takes the form

yg_{0}(x, y)(x + h.o.t.) = 0 or g_{0}(x, y)(x + h.o.t.) = 0.

Since D meets each of the other branches generically, its equation can be specialized to x = 0 without changing the topological type of C_{0}. □

Essentially the same proof yields the following corollary:

Corollary 2.9 Let C_{0} be of toric or quasitoric type. Then there exist local coordinates x and y and positive integers p and q such that C_{0} has one of the following analytic equations

1. g_{0}(x, y) + h.o.t. = 0, with p ≥ q > 1;

2. x(g_{0}(x, y) + h.o.t.) = 0, with p > q > 1;

3. y(g_{0}(x, y) + h.o.t.) = 0, with p > q > 1 and q |̸p;

4. xy(g_{0}(x, y) + h.o.t.) = 0, with p > q > 1 and q |̸p;

where g_{0}(x, y) = x^{p} + … + y^{q} is weighted-homogeneous and square-free. (In particular, C_{0} is semiquasihomogeneous.)

The weights on x and y are the same as in Proposition 2.8. Of course, ‘higher order’ means higher degree with respect to this grading.

The equations of Proposition 2.8 all define singularities of toric or quasitoric type. We therefore obtain the following corollary:

Corollary 2.10 Each isolated plane curve singularity satisfying the conditions of Proposition 2.5 (resp. 2.6) is of toric (resp. quasitoric) type.

Note also that Proposition 2.8 implies that each unibranch singularity of toric type admits an equisingular specialization to x^{p} = y^{q}. This is a well known result of Zariski (chapter III §2.3 of [25]), and serves as the foundation for the construction of moduli spaces for unibranch singularities of toric type. For a systematic study of parameter/moduli spaces for unibranch curve singularities, see the survey [11] and the lecture notes of Zariski (with appendix by Teissier) [25].

We will also use the following result of Wahl [22] on equisingular deformation spaces:

Theorem 2.11 Let C_{0} be an isolated plane curve singularity. Then the equisingular deformation space of C_{0} is smooth. For the singularity x^{p} = y^{q} it may be represented as

y^{q} = x^{p} + ∑ t_{ij}x^{i}y^{j}

where 0 ≤ i ≤ p − 2, 0 ≤ j ≤ q − 2, and qi + pj ≥ pq.

By Proposition 2.8, we may regard the singularities appearing in this family as a parameter space for the singularities of type (p, q). See the papers [11] and [19] for an elaboration of this point of view. With this as motivation, we make the following definition:

Definition 2.12 The codimension c(p, q) of the singularities of type (p, q) is the codimension of the family in Theorem 2.11 in the versal deformation space of y^{q} = x^{p}.

This will be computed in section 7.

3 Basic properties of local stable reduction

The stable reduction theorem for curves was proved by Deligne and Mumford [6]. Another proof may be found in Artin and Winters [3], and a good general introduction is the recent book of Harris and Morrison [12], chapter 3C.

Let C_{0} be the germ of an isolated plane curve singularity. Set Δ = Spec ℂ[[t]] or {t : |t| < 1}. Let π : C → Δ be a smoothing of C_{0}, i.e. π is flat, C_{0} = π^{−1}(0), and C_{t} := π^{−1}(t) is smooth for t ≠ 0. Local stable reduction is a procedure for obtaining a family of nodal curves π^{c} : C^{c} → Δ̃. It involves the following steps:

1. Carry out semistable reduction, following [15]. We find a base change Δ̃ → Δ and a resolution ρ : C̃ → C ×_{Δ} Δ̃ such that the central fiber C̃_{0} is a reduced normal crossings divisor.

2. Obtain C^{c} by taking the log canonical model of (C̃, C̃_{0}) relative to the morphism ρ.

We use a relative version of the minimal model program, i.e., we take minimal models relative to a proper morphism of normal varieties. A good exposition may be found in [14].

Over the punctured disc Δ̃^{∗}, C^{c} coincides with the pull-back of the original family. The central fiber C_{0}^{c} = C_{1} ∪ C_{T} where C_{1} is the normalization of C_{0} and C_{T} = C_{0}^{c} − C_{1}. (We are assuming C_{0} is not a node.) Let C_{1} ∩ C_{T} = {p_{1}, …, p_{b}}, where b is the number of branches of C_{0}. Note that the pointed curve (C_{1}, p_{1}, …, p_{b}) depends only on C_{0}, and not on the choice of smoothing.

Definition 3.1 The pointed curve (C_{T} , p_{1}, …, p_{b}) is called the tail of the local stable reduction.

Proposition 3.2 Let C_{0} be an isolated plane curve singularity and let T_{C_{0}} denote all the tails obtained from various smoothings of C_{0}. These tails are pointed stable curves and T_{C_{0}} ⊂ \overline{M}_{γ,b} is a connected closed subvariety of the moduli space.

Let C → Δ be a smoothing of C_{0}. After any base change t = u^{N}, the family C ×_{Δ} Δ̃ remains normal. If C^{c} is the stable reduction, then we have a morphism of normal varieties ϕ : C^{c} → C ×_{Δ} Δ̃. The tail C_{T} is the fiber over the point x = y = u = 0 so it is connected by Zariski’s Main Theorem. Since (C^{c}, C_{0}^{c}) is the log canonical model of C ×_{Δ} Δ̃ relative to ρ, it follows that C_{T} is nodal and ω_{C_{T}} (p_{1} + … p_{b}) is ample. In particular, (C_{T} , p_{1}, …, p_{b}) is a pointed stable curve.

We now show that T_{C_{0}} is closed in the corresponding moduli space of stable pointed curves. Let Ver_{C_{0}} be the versal deformation space of C_{0} and let Λ be a linear series globalizing Ver_{C_{0}}, i.e. some member L_{0} ∈ Λ has a single singularity p analytically equivalent to C_{0}, and the induced analytic map Λ̂_{L_{0}} → Ver_{C_{0}} is an isomorphism. For instance, we may take Λ to be a linear system of plane curves with sufficiently large degree. If the generic member of Λ has genus g, we obtain a rational map Λ −−→ \overline{M}_{g}. This map has indeterminacy at the point L_{0} and the total transform of L_{0} is a closed subvariety W of the moduli space. Since (C_{1}, p_{1}, …, p_{b}) depends only on L_{0}, we find that W ≃ T_{C_{0}}. □

We want to describe T_{C_{0}} explicitly for certain singularities C_{0}. To do this, we consider pairs (S_{0}, C_{0}) where S_{0} = Spec ℂ[[x, y]] and C_{0} ⊂ S_{0}. We consider the smoothing C → Δ as a subscheme of the threefold S := S_{0} × Δ. Local stable reduction for the family of pairs π : (S, C) → Δ involves the following steps:

1. Carry out semistable reduction, following [15] and [5] Theorem 7.17. We find a base change Δ̃ → Δ and a resolution

ρ : (S̃, C̃) → (S ×_{Δ} Δ̃, C ×_{Δ} Δ̃)

such that the central fiber S̃_{0} is reduced and C̃∪S̃_{0} is a normal crossings divisor.

2. Obtain (S^{c}, C^{c}) by taking the log canonical model of (S̃, C̃+S̃_{0}) relative to the morphism ρ.

The central fiber (S_{0}^{c}, C_{0}^{c}) is called the local stable limit. Local stable reductions satisfy the following:

Proposition 3.3 C_{0}^{c} is a Weil divisor, (S_{0}^{c}, C_{0}^{c}) is semilog canonical, and hence C_{0}^{c} is nodal. Furthermore, K_{S^{c}} + C^{c} is Cartier along C^{c} and restricts to K_{C^{c}}.

For the convenience of the reader we sketch the proof (see also §2 of [1].) The pair (S^{c}, C^{c} + S_{0}^{c}) is log canonical so C_{0}^{c} is generically reduced. Since the local stable reduction agrees with our original family away from the central fiber, the exceptional divisors with log discrepancy < 1 are supported in the central fiber. We obtain that (S^{c}, C^{c}) is canonical hence S^{c} (and also S_{0}^{c}) is Cohen-Macaulay. Furthermore, C^{c} is also Cohen-Macaulay [17] 17.5 and thus C_{0}^{c} has no imbedded points. Hence we may apply adjunction ([17] 17.2) to obtain that (S_{0}^{c}, C_{0}^{c}) is semilog canonical. The fact that C_{0}^{c} is nodal follows from the classification of semilog canonical singularities [17] 12.2.4.

Now K_{S^{c}} + C^{c} is Cartier along C^{c} provided that K_{S_{0}^{c}} + C_{0}^{c} is Cartier along C_{0}^{c}. (The log canonical bundle necessarily restricts well to the central fiber.) Assume that K_{S_{0}^{c}} + C_{0}^{c} is not Cartier at p ∈ C_{0}^{c}, and let r > 1 be the corresponding index. Restricting to a small analytic neighborhood about p, we can take an index-one cover i : (T, D) → (S_{0}^{c}, C_{0}^{c}). Note that i is an r-fold cover totally ramified at p and K_{T} + D is Cartier. It follows that i^{∗}K_{C} = K_{D}(−(r − 1)p), (K_{T} + D)|D = K_{D}, and hence K_{S_{0}^{c}} + C_{0}^{c} (when restricted to C_{0}^{c}) has different \frac{r−1}{r} at p ([17] 16.6). This contradicts the flatness of C_{0}^{c} → Δ̃. The isomorphism (K_{S^{c}} + C^{c})|C^{c} = K_{C^{c}} follows by the adjunction formula ([17] 16.4.2).□

Corollary 3.4 The boundary C^{c} coincides with the local stable reduction of C → Δ described above.

We now set some notation. The central fiber S_{0}^{c} = S_{1} ∪ S_{T} where S_{1} is the proper transform of S_{0} and S_{T} the closure of S_{0}^{c} − S_{1}. We write S_{T} = S_{2} ∪ … ∪ S_{n} where the S_{j} are irreducible, and C_{j} = S_{j} ∩ C_{0}^{c} for the part of the tail C_{T} lying on S_{j}; it is possible that C_{j} = ∅ for some j. We also write B_{j} = S_{j} ∩ \overline{S_{0}^{c} − S_{j}} and B_{T} = S_{T} ∩ S_{1}; note that B_{T} = B_{1}.

Definition 3.5 The log surface (S_{T} , C_{T} + B_{T}) is called the surface tail of the local stable reduction.

Proposition 3.6 Let C_{0} be an isolated plane curve singularity and let T′_{C_{0}} denote all the surface tails obtained from various smoothings of C_{0}. These tails are stable log surfaces and form a connected closed subvariety of the appropriate moduli space. There is a natural surjective forgetting morphism T′_{C_{0}} → T_{C_{0}}.

This proof follows the one for Proposition 3.2, using basic properties of stable reduction for log surfaces. We shall prove that (S_{1}, B_{1} +C_{1}) depends only on C_{0} and not on the choice of smoothing in Theorem 4.1. Since ϕ : S^{c} → S×_{Δ}Δ̃ is projective and birational, it is a blow-up of a suitable ideal sheaf. The support of the exceptional divisor is S_{T}. Because S^{c} is Cohen-Macaulay (Proposition 3.3), the exceptional divisor (a Cartier divisor) is also Cohen-Macaulay, as is S_{T}. The pair (S_{T} , C_{T} + B_{T}) is therefore semilog canonical. The log canonical bundle K_{S^{c}} + C^{c} is ample relative to ϕ. By the adjunction formula ([17] 16.4.3) (K_{S^{c}} + C^{c})|S_{T} = K_{S_{T}} + C_{T} + B_{T}, so (S_{T} , C_{T} + B_{T}) is stable. This establishes the first claim. The second claim is proved precisely as in Proposition 3.2. The existence of the forgetting morphism follows from Corollary 3.4. □ Remark: Can the forgetting map T′_{C_{0}} → T_{C_{0}} have positive dimensional fibers?

4 Computing local stable reductions

4.1 The Main Component Theorem

We give a simple recipe for computing the component S_{1} of the local stable reduction:

Theorem 4.1 (Main Component Theorem) Let C_{0} ⊂ S_{0} be an isolated plane curve singularity, (S_{0}^{c}, C_{0}^{c}) the local stable limit of some smoothing of C_{0}, S_{1} ⊂ S_{0}^{c} the proper transform of S_{0}, B_{1} = S_{1}∩\overline{S_{0}^{c} − S_{1}}, and C_{1} = C_{0}^{c}∩S_{1}. There is a birational morphism σ : S_{1} → S_{0} such that the divisor K_{S_{1}}+B_{1}+C_{1} is ample relative to σ. Indeed, (S_{1}, B_{1} + C_{1}) is obtained from an imbedded resolution r : P_{1} → S_{0} by taking the log canonical model of (P_{1}, E + C_{1}) relative to r.

The minimal model program gives us an induced map ϕ : S^{c} → S ×_{Δ} Δ̃ which is an isomorphism except over the origin 0 = (x = y = t = 0). The exceptional divisors of ϕ correspond to S_{0}^{c}\S_{1}, and ϕ restricts to a birational morphism σ : S_{1} → S_{0}. By the adjunction formula ([17] ch. 16.4.3), we have

K_{S_{1}} + B_{1} + C_{1} = (K_{S^{c}} + S_{0}^{c} + C^{c})|S_{1}

which is σ-ample because (S^{c}, C^{c}) is the log canonical model.

To prove the second part of the theorem, we must verify that B_{1} equals the image of the divisor E defined above. In other words, we must show that every σ-exceptional divisor is contained in some ϕ-exceptional divisor. Assume this is not the case, so the image of some component E′ ⊂ E in S_{1} is not contained in any of the ϕ-exceptional divisors. Then at the generic point of image of E′, which is a smooth point of S^{c}, ϕ has one-dimensional exceptional locus. However S ×_{Δ} Δ̃ is also smooth, and the exceptional locus of any birational morphism of smooth varieties has pure codimension one. □

Generally, the map σ : S_{1} → S_{0} can be expressed as the blow-up of a natural ideal sheaf. Consider the log canonical divisor

D :≡ h^{∗}(K_{S_{1}} + C_{1} + B_{1}) = K_{P_{1}} + C_{1} + ∑_{i=1}^{n} a_{i}E_{i}

on the resolution h : P_{1} → S_{1}. Let N be the smallest positive number such that ND is integral, globally generated relative to r, and generates ⊕_{m≥0}r_{∗}O_{P_{1}}(mND) as an O_{S_{0}} algebra. (In practice, it often suffices to assume that ND is integral.) The log adjoint ideal A_{C_{0}} is defined as the push-forward r_{∗}O_{P_{1}}(ND) ⊂ O_{S_{0}}. This can interpretted geometrically: the images in S_{0} of the members of |ND| yield a linear series on S_{0}, the log adjoint series. The base locus of the log adjoint series is the log adjoint ideal. Using basic properties of blowing up, we obtain the following:

Proposition 4.2 Retain the notation of Theorem 4.1. Then σ : S_{1} → S_{0} coincides with the blow-up of S_{0} along the log adjoint ideal A_{C_{0}}.

Finally, we should point out when local stable reductions coincide with global stable reductions. Assume we are given a family of log surfaces (S, C) → Δ, such that the fibers over the punctured disc Δ^{∗} are smooth and stable. In particular, K_{S_{t}} + C_{t} is ample for t ≠ 0. Furthermore, assume that S_{0} is smooth but that C_{0} has an isolated singularity. We can apply local stable reduction to obtain a family (S^{c}, C^{c}) → Δ̃ dominating (S ×_{Δ} Δ̃, C ×_{Δ} Δ̃), so that the central fiber (S_{0}^{c}, C_{0}^{c}) has semilog canonical singularities. When is this also the global stable reduction? Of course, (S_{0}^{c}, C_{0}^{c}) is semilog canonical and global stable reductions are unique, so it suffices to check whether K_{S_{0}^{c}} + C_{0}^{c} is ample. This is clearly valid for each irreducible component of S_{0}^{c} that is exceptional for the map ϕ : S^{c} → S ×_{Δ} Δ̃. It follows that the local stable reduction is the global stable reduction whenever K_{S_{1}} + B_{1} + C_{1} is ample (using the notation of Theorem 4.1).

4.2 Computing log canonical models

We recall some results on surface singularities. Consider the cyclic quotient surface singularity arising from the group action

(x, z) → (ϵ^{a}x, ϵz)

where ϵ is a primitive rth root of unity, 1 ≤ a < r, and (a, r) = 1. The minimal resolution of this singularity can be described quite explicitly. It consists of a chain of rational curves E_{1}, E_{2}, …, E_{n} with self-intersections E_{i}^{2} = −b_{i}

∘^{−b_{1}} − ⋯ − ∘^{−b_{n}} .

The b_{i} are computed from the continued fraction representation

\frac{r}{a} = b_{1} − \frac{1}{b_{2} − \frac{1}{b_{3}−...}} .

The proper transforms of x = 0 and z = 0 meet the first and last exceptional curves of this chain. (See [10] §2.6 for a good exposition of this subject.) We shall use the fraction \frac{r}{a} to label the cyclic quotient. For instance, the singularity A_{g} corresponds to the fraction \frac{g+1}{g}. This notation depends on the choice of one of two possible orderings for the exceptional divisors of the minimal resolution.

We now describe the log minimal and log canonical models associated to singularities of toric type.

Proposition 4.3 Retain the notation of Theorem 4.1, and assume further that C_{0} is of type (p, q). Then S_{1} is obtained from P_{1} by contracting all the exceptional curves disjoint from C_{1}, and B_{1} is the image of the unique exceptional curve meeting C_{1}. In particular, we have

1. If p = q > 2 then S_{1} is smooth and equal to the minimal imbedded resolution of C_{0}.

Figure 3: Log canonical model of y^{4} = x^{10}

2. If q|p but q ≠ p, then S_{1} has one cyclic quotient singularity along B_{1} (but disjoint from C_{1}).

3. If q |̸p, then S_{1} has two cyclic quotient singularities along B_{1} (but disjoint from C_{1}).

Finally, (S_{1}, C_{1} + B_{1}) has log terminal singularities and thus coincides with the log minimal model (except in the case where p = q = 2).

See figure 3 for a schematic representation of such a log canonical model. The number in parentheses is the self-intersection of B_{1} in the minimal resolution of S_{1}. The fraction denotes the corresponding quotient singularity.

We assume that C_{0} is not an ordinary node; if C_{0} is a node then the proposition is evident. Here we use Proposition 2.5 and Lemma 2.7. Let E_{n} be the unique exceptional curve meeting C_{1}. We claim that every exceptional curve besides E_{n} is contracted when we take the log minimal model; B_{1} corresponds to the image of E_{n}. Let F be the union of the exceptional curves besides E_{n}; it has zero (resp. one, resp. two) connected components if q = p (resp. q|p, resp. q |̸p). Each such component is a chain of exceptional curves, with one curve at the end of the chain intersecting E_{n} once. Such chains are log terminal [17] ch.3, so they are necessarily contracted when we take the log minimal model. As we have seen, contracting such a chain of rational curves yields a cyclic quotient singularity.

On the other hand, E_{n} may not be contracted even after taking the log canonical model. If we contracted E_{n}, then the boundary would have singularities worse than normal crossings, which contradicts the fact that the singularities are log canonical([17] chapter 3). □

Similar results hold for singularities of quasitoric type:

Proposition 4.4 Retain the notation of Theorem 4.1, and assume further that C_{0} is of quasitoric type. Then S_{1} is obtained from P_{1} by contracting

Figure 4: Toroidal representation of the log canonical model S_{1}

all the exceptional curves disjoint from the nondistinguished branches of C_{1}; B_{1} is the image of the unique exceptional curve meeting those branches. S_{1} has (at most) two cyclic quotient singularities along B_{1}; each distinguished branch passes through one of these quotient singularties.

The proof uses Proposition 2.6 and the classification of log canonical singularities. Remark: For singularities of toric and quasitoric type σ : S_{1} → S_{0} has a simple toroidal interpretation. Choose coordinates x and y and integers p and q satisfying the conclusions of Corollary 2.9. Let b = gcd(p, q) and consider the ideal A generated by the weighted monomials {x^{i}y^{j} : pj + qi ≥ \frac{pq}{b}\}. Blowing up A yields a resolution of C_{0}, imbedded in a normal surface with cyclic quotient singularities. This corresponds to the toric blow-up represented by the fan depicted in figure 4. The horizontal and vertical vectors in the fan correspond to the proper transforms of y = 0 and x = 0 respectively. The propositions above imply that this toroidal blow-up equals S_{1}, which gives a combinatorial method for describing the singularities of S_{1} (see [10] §2.6). Finally, we should point out that A is precisely the log adjoint ideal A_{C_{0}} defined in the previous section.

5 Constructions of limiting surfaces

5.1 Geometry of weighted projective spaces

We recall some basic facts about the weighted projective plane ℙ(r, s, 1) (see [7] for a comprehensive survey). It may be represented as the toric surface obtained from the complete fan with one-dimensional faces (1, 0), (0, 1), and (−r, −s) [10] §2.2. In particular, ℙ(r, s, 1) is isomorphic to ℙ(nr, ns, 1), so we assume that r and s are relatively prime and r ≥ s. Let B_{2} denote the distinguished divisor corresponding to the face (−r, −s), and let D_{1} and D_{2} be the divisors corresponding to (1, 0) and (0, 1). These satisfy the rational equivalences rB_{2} ≡ D_{1} and sB_{2} ≡ D_{2}; the intersection form is

\begin{aligned}& D_{1} & D_{2} & B_{2} \\ D_{1} & r/s & 1 & 1/s \\ D_{2} & 1 & s/r & 1/r \\ B_{2} & 1/s & 1/r & 1/rs \end{aligned} .

The Picard group is generated by L = rsB_{2} ([10] 3.3) and the canonical divisor K_{ℙ(r,s,1)} = −(B_{2} + D_{1} + D_{2}) ([10] 4.4).

If r > s = 1 (resp. s > 1) then ℙ(r, s, 1) has one (resp. two) cyclic quotient singularities, occuring at the intersection D_{2} ∩ B_{2} (resp. at the intersections D_{2} ∩B_{2} and D_{1} ∩B_{2}). We should point out that (ℙ(r, s, 1), B_{2} + D_{1} + D_{2}) is always log canonical.

We describe some natural ‘linear series’ on ℙ(r, s, 1), i.e. spaces of sections of O_{ℙ(r,s,1)}(nB_{2}) for n ≥ 0. Of course, these are not generally invertible sheaves, but rather rank-one reflexive sheaves. The dimensions of their global sections are determined by the isomorphism of graded rings [7] 1.4

Proj ℂ[x^{r}, y^{s}, z] ≅ ⊕_{n≥0}H^{0}(O_{ℙ(r,s,1)}(nB_{2})).  (1)

If r > s then h^{0}(O_{ℙ(r,s,1)}(D_{2})) = 2 and h^{0}(O_{ℙ(r,s,1)}(D_{1})) = ⌊\frac{r}{s}⌋ + 2. If s = 1 then ℙ(r, 1, 1) is the cone over a rational normal curve of degree r in ℙ^{r}; B_{2} ≡ D_{2} is the class of a ruling and D_{1} is the class of a hyperplane section. If s > 1 then ℙ(r, s, 1) is still ruled by the divisors linearly equivalent to D_{2}.

For each positive integer b, the Cartier divisor bL is very ample and its higher cohomologies vanish ([10] pp. 70-74). It follows that the general member of |bL| is smooth of genus \frac{b}{2}(brs − r − s − 1) + 1, contained in the smooth locus of ℙ(r, s, 1), and h^{0}(bL) = \frac{b}{2}(brs + r + s + 1) + 1. Finally, |bL| maps B_{2} to a rational normal curve of degree b. Thus there is a smooth member of |bL| passing through any collection of b smooth points of ℙ(r, s, 1) lying on B_{2}.

If r > s then a generic Weil divisor C_{2} ∈ |bL + D_{2}| is smooth of genus \frac{b}{2}(brs+s−r−1), but it contains the singularity at B_{2}∩D_{2}. In the local divisor class group at B_{2} ∩ D_{2} C_{2} is linearly equivalent to D_{2}, so (ℙ(r, s, 1), C_{2} + B_{2}) is log canonical. If s > 1 then the generic member C_{2} ∈ |bL + D_{1}| is smooth of genus \frac{b}{2}(brs − s + r − 1), but it contains the singularity at B_{2} ∩ D_{1}; again (ℙ(r, s, 1), C_{2} + B_{2}) is log canonical. Similarly, the generic member C_{2} ∈ |bL + D_{1} + D_{2}| is smooth of genus \frac{b}{2}(brs + r + s − 1), contains both singularities B_{2} ∩ D_{1} and B_{2} ∩ D_{2}, and (ℙ(r, s, 1), C_{2} + B_{2}) is log canonical.

We will use the following result for dimension counts later in this paper. It is a consequence of the cohomology computations of [7] §2, or of the description of the automorphisms of ℙ(r, s, 1) in terms of the automorphisms of the corresponding graded algebras.

Proposition 5.1 Let r and s be relatively prime integers with r ≥ s > 0. Then the automorphism group of ℙ(r, s, 1) has dimension ⌊\frac{r}{s}⌋ + 4 if s ≠ 1, dimension r + 5 if s = 1 and r ≠ 1, and dimension 8 if r = s = 1.

5.2 Construction of degenerations

In this section, we construct certain surfaces S_{0}^{c} containing tails arising from plane curve singularities C_{0} of toric and quasitoric type. We describe how these surfaces are obtained as central fibers of birational modifications of the trivial families S = S_{0} × Δ.

Theorem 5.2 Let C_{0} ⊂ S_{0} be a singularity toric or quasitoric type which is not an ordinary node. There exists a surface S_{0}^{c} with the following properties

1. S_{0}^{c} has two irreducible components S_{1} and S_{2} and has semilog terminal singularities.

2. S_{1} is the log canonical model of the imbedded resolution of C_{0} (cf. Theorem 4.1).

3. S_{2} is isomorphic to a weighted projective space ℙ(p, q, 1).

4. S_{0}^{c} can be smoothed to S_{0} so that the total space S^{c} has terminal singularities.

Figure 5: The toroidal blow-up ϕ : S^{c} → S

Choose analytic coordinates x and y and integers p and q satisfying the conclusions of Corollary 2.9. We assume that p ≥ q and set b = gcd(p, q). Consider the ideal I^{c} ⊂ O_{S} generated by t and generators of the log adjoint ideal A_{C_{0}}. By the remark at the end of section 4.2, this may be represented torically as \{t, x^{i}y^{j} : pj + qi ≥ \frac{pq}{b}\}. Let ϕ : S^{c} → S be the blow-up of S along I^{c}. We express this torically as follows: represent S by a fan with one cone generated by the vectors v_{1} = (1, 0, 0), v_{2} = (0, 1, 0), and v_{3} = (0, 0, 1), corresponding to the divisors y = 0, x = 0, and t = 0. Then ϕ corresponds to the toric blow-up represented by a fan with three cones, each spanned by two of the v_{i} along with the vector (\frac{p}{b}, \frac{q}{b}, 1). (See figure 5.)

Basic properties of divisors in toric varieties imply that the central fiber S_{0}^{c} ⊂ S^{c} (defined by t = 0) is reduced. By the description of orbit closures in [10] 3.1, it consists of two components isomorphic to the toric blow-up S_{1} → S_{0} and the weighted projective space S_{2} = ℙ(p/b, q/b, 1) respectively. Each of these surfaces has (up to) two quotient singularities, lying along their common intersection B_{1} = B_{2}. Away from these quotient singularities, S_{1} and S_{2} intersect in normal crossings.

We claim the threefold S^{c} has terminal singularities; this implies (by adjunction) that S_{0}^{c} is semilog terminal. It suffices to prove that an affine toric threefold U with fan generated by (1, 0, 0), (0, 1, 0), (1, s, r) (with r, s relatively prime) has terminal singularities. Each affine neighborhood of S^{c} has this form after an appropriate permutation. We represent this as a quotient singularity of index r. Set w_{1} = (1, 0, 0), w_{2} = (0, 1, 0), w_{3} = (1, s, r) and let N_{1} be the corresponding lattice spanned by these elements. Now N_{1} is an index r subgroup of the full torus N and

N = N_{1} + \frac{ℤ}{r}(w_{3} − w_{1} − rw_{2}) = N_{1} + \frac{ℤ}{r}((r − 1)w_{1} + (mr − s))w_{2} + w_{3})

where m ∈ ℕ and 0 < mr−s < r. The toric variety with torus corresponding to N_{1} and fan consisting of the cone generated by w_{1}, w_{2}, w_{3} is just 𝔸^{3}. The toric variety U ≅ 𝔸_{3}/(ℤ/rℤ), where ℤ/rℤ acts with weights (r−1, mr−s, 1). It follows that U has terminal singularities [20]. □

6 Description of the tails

6.1 General theorems

Let C_{0} ⊂ S_{0} be a singularity of toric or quasitoric type which is not an ordinary node, S_{0}^{c} the surface constructed in Theorem 5.2, and C_{1} ⊂ S_{1} the curve constructed in Theorem 4.1. Let C_{2} ⊂ S_{2} be a Weil divisor in the linear series

1. |O_{ℙ(p/b,q/b,1)}(bL)| if C_{0} is of type (p, q);

2. |O_{ℙ(p/b,q/b,1)}(bL + D_{2})| if C_{0} is quasitoric of type x(x^{p} − y^{q}) = 0 (with p > q > 1);

3. |O_{ℙ(p/b,q/b,1)}(bL + D_{1})| (resp. |O_{ℙ(p/b,q/b,1)}(bL + D_{1} + D_{2})|) if C_{0} is quasitoric of type y(x^{p} − y^{q}) = 0 (resp. xy(x^{p} − y^{q}) = 0) (with p > q > 1 and q |̸p).

See section 5.1 for notation; we use b = gcd(p, q), r = \frac{p}{b}, and s = \frac{q}{b} throughout. Furthermore, assume that C_{2} is nodal and C_{1} ∩ B_{1} = C_{2} ∩ B_{2}. Let C_{0}^{c} ⊂ S_{0}^{c} be the Weil divisor consisting of the union of C_{1} and C_{2}. We should emphasize that C_{0}^{c} is Cartier iff C_{0} is of toric type.

Theorem 6.1 Retain the notation of the paragraph above and let S^{c} → Δ be the smoothing of S_{0}^{c} constructed in Theorem 5.2. Then there exists a Weil divisor C^{c} ⊂ S^{c} such that C^{c}|S_{0}^{c} = C_{0}^{c} and C_{t}^{c} is smooth for t ≠ 0. Furthermore, (S^{c}, C^{c}) is the local stable reduction of a smoothing of C_{0}.

We first claim each rank-one reflexive sheaf F on S^{c} restricts to a depth-two sheaf on the central fiber. In the proof of Theorem 5.2, we saw that S^{c} is covered by affines U which are quotients of 𝔸^{3} by ℤ/rℤ acting with weights (r−1, mr−s, 1). In other words, U is the cone over some weighted projective space ℙ(r − 1, mr − s, 1). Each divisor class on U may be represented as the cone over a divisor of the weighted projective space, and the corresponding reflexive sheaves take the form O_{U}(nCone(B_{2})) for some n. Using standard results relating the local cohomology of a graded module to the ordinary cohomology of its projectivization [8] Theorem A.4.1, we find that

H_{0}^{i}(O_{U}(nCone(B_{2}))) = H^{i−1}(O_{ℙ(r−1,mr−s,1)}(nB_{2}))

for i > 1. By standard cohomology computations for weighted projective spaces [7] 1.4, we conclude that the local cohomology vanishes for i ≤ 3. Consequently, reflexive rank-one sheaves on S^{c} have depth three and restrict to depth-two sheaves on the central fiber.

The toric representation of S^{c} shows that S_{1} defines a Weil divisor generating the divisor class group of S^{c}. Recall that S_{1} ∩ S_{2} = B_{1} = B_{2}. If C_{0} is of toric type (p, q), then the line bundle O_{S_{0}^{c}}(C_{0}^{c}) is the restriction of F = O_{S^{c}}(\frac{pq}{b}S_{1}) to the central fiber. More generally, if C_{0} is of quasitoric type and C_{2} is a member of |O_{ℙ(p/b,q/b,1)}(bL + D_{2})| (resp. |O_{ℙ(p/b,q/b,1)}(bL + D_{1})|, |O_{ℙ(p/b,q/b,1)}(bL + D_{1} + D_{2})|) then O_{S_{0}^{c}}(C_{0}^{c}) is the restriction of F = O_{S^{c}}((\frac{pq+q}{b}S_{1}) (resp. O_{S^{c}}((\frac{pq+p}{b}S_{1}), O_{S^{c}}((\frac{(pq+p+q)}{b}S_{1})).

The divisors O_{S_{2}}(C_{2}) have no higher cohomology by [7] 1.4. This is also the case for O_{S_{1}}(C_{1}), as can be seen by applying a variant of Kawamata-Viehweg vanishing [14] 1.2.5. It follows that O_{S_{0}^{c}}(C_{0}^{c}) also has no higher cohomology. In particular, each section of this reflexive sheaf is the restriction of a section of a suitable F. We choose C^{c} to be a general section of F that restricts to C_{0}^{c} along the central fiber. The fibers C_{t}^{c} are smooth for t ≠ 0.

We check that (S^{c}, C^{c}) is a local stable reduction. We know that (S_{0}^{c}, C_{0}^{c}) is semilog canonical, K_{S_{0}^{c}} + C_{0}^{c} is relatively ample, and S^{c} is ℚ-factorial, so inversion of adjunction ([17] 17.12) gives that (S^{c}, C^{c} + S_{0}^{c}) is log canonical. The uniqueness of log canonical models implies this is a local stable reduction. □

We apply this to describe certain smooth tails of singularities of toric type. The statement is simplest in the unibranch case.

Theorem 6.2 (Main Theorem 1) Let C_{0} be a singularity of type (p, q), where p and q are relatively prime and p > q. Let C_{2} be a nodal curve in the linear series O_{ℙ(p,q,1)}(L) and set p_{1} = B_{2} ∩ C_{2}. Then we have

(ℙ(p, q, 1), C_{2} + B_{2}) ∈ T′_{C_{0}} and (C_{2}, p_{1}) ∈ T_{C_{0}}.

Furthermore, these curves satisfy the following properties:

1. p_{1} is a subcanonical point of C_{2}, i.e. (pq − p − q − 1)p_{1} = K_{C_{2}}.

2. C_{2} is q-gonal, with g_{q}^{1} equal to |H^{0}(O_{C_{2}}(qp_{1}))|.

3. More generally, for n = 1, …, pq − p − q − 1

h^{0}(O_{C_{2}}(np_{1})) = {(i, j) : i, j ≥ 0 and qi + pj ≤ n}.

Recall that L is the effective generator of the Picard group of ℙ(p, q, 1). The first assertion follows immediately from Theorem 6.1 and the definitions of T_{C_{0}} and T′_{C_{0}}. The geometric statements about C_{2} are consequences of the results on weighted projective spaces in § 5.1. The first statement follows from the fact that the divisor class group of ℙ(p, q, 1) is generated by B_{2}, which meets C_{2} at the point p_{1}. The second statement follows from our description of the sections of O_{ℙ(p,q,1)}(qB_{2}), which restrict to give the g_{q}^{1}. Geometrically, the ruling of ℙ(p, q, 1) restricts to the g_{q}^{1} on C_{2}. As for the third statement, we first observe that the restriction

H^{0}(O_{ℙ(p,q,1)}(nB_{2})) → H^{0}(O_{C_{2}}(np_{2}))

is surjective for all n [7] 1.4. The isomorphism of graded rings in equation 1 gives the desired formula. □ Remark: In the special case where the singularity is analytically isomorphic to y^{q} = x^{p}, this result can be deduced from Pinkham’s work (see [21] §1.16 and 13). More generally, he considers certain smoothings of curve singularities with ℂ^{∗}-action, and shows that curves C_{2} with distinguished Weierstrass point p_{1} naturally arise as tails. Weierstrass points also play an important role in the theory of ‘limit linear series’ of Eisenbud and Harris [9].

In the case where the singularity is not unibranch, we have to keep track of a compatibility condition along the intersection of surfaces S_{1} and S_{2}. This makes our statements a bit more complicated.

Theorem 6.3 (Main Theorem 2) Let C_{0} be a singularity of type (p, q) with b = gcd(p, q). Let (S_{1}, C_{1} + B_{1}) be the surface obtained from Theorem 4.1, and let S_{0}^{c} be the surface obtained by gluing S_{1} and S_{2} as in Theorem 5.2. Let C_{2} be a nodal curve in the linear series O_{ℙ(p/b,q/b,1)}(bL), such that C_{2} ∩ B_{2} = {p_{1}, p_{2}, …, p_{b}} = C_{1} ∩ B_{1}. Then we have

(ℙ(p, q, 1), C_{2} + B_{2}) ∈ T′_{C_{0}} and (C_{2}, p_{1}, p_{2}, …, p_{b}) ∈ T_{C_{0}}.

Furthermore, these tails satisfy the following properties:

1. p_{1} + p_{2} + … + p_{b} is a subcanonical divisor of C_{2}, i.e.

(\frac{pq}{b} − \frac{p}{b} − \frac{q}{b} − 1)(p_{1} + p_{2} + … + p_{b}) = K_{C_{2}}.

2. C_{2} is q-gonal, with g_{q}^{1} = H^{0}(O_{C_{2}}(\frac{q}{b}(p_{1} + p_{2} + … + p_{b}))).

3. More generally, for n = 1, …, \frac{pq}{b} − \frac{p}{b} − \frac{q}{b} − 1

h^{0}(O_{C_{2}}(n(p_{1} + … + p_{b}))) = {(i, j) : i, j ≥ 0 and qi + pj ≤ nb}.

Again, this follows from Theorem 6.1 and the results on weighted projective spaces.

Proposition 6.4 For each singularity C_{0} of toric type, Theorem 6.2 (or 6.3) yields an irreducible component of T′_{C_{0}}.

We claim that the space of allowable deformations of S_{0}^{c} is smooth of dimension one, and thus consists only of smoothings to S_{0}. By definition, a deformation is allowable if it lifts locally to index-one covers. First, it is not hard to check that weighted projective spaces admit no equisingular deformations, hence S_{0}^{c} also admits no equisingular deformations. On the other hand, the results from §3.2 of [13] imply that the versal deformation space of the singularities of S_{0}^{c} is smooth of dimension one. □

Proposition 6.4 suggests that the tails described in Theorems 6.2 and 6.3 might also yield an irreducible component of T_{C_{0}}. Proposition 7.3 provides further evidence for this contention. We therefore make the following conjecture:

Conjecture 6.5 Let C_{0} be a singularity of toric type. Then the tails described in Theorems 6.2 and 6.3 yield an irreducible component of T_{C_{0}}.

We also raise the following wildly speculative conjecture:

Conjecture 6.6 Let C_{0} be a (simple) singularity of toric (or even quasitoric) type. Then T_{C_{0}} is irreducible.

This is known for singularities of multiplicity two and some singularities of multiplicity three. The proof involves analyzing the allowable deformation spaces of each of the possible degenerate surface tails.

One can formulate analogous results for singularities of quasitoric type. Again, all the weighted plane curves in a given divisor class (satisfying certain compatibility conditions) appear as tails of stable reductions. We work out certain examples in subsection 6.3.

Figure 6: Some local stable reductions for y^{2} = x^{2g+2}

6.2 Concrete examples

We illustrate these results by describing curves arising as tails for some simple singularities.

6.2.1 A_{2g+1} singularities

See figure 6 for a schematic representing this case; this figure uses the notational conventions of figure 3. Our general analysis implies that (C_{2}, p_{1} + p_{2}) is a hyperelliptic curve of genus g, where p_{1} and p_{2} are exchanged under the action of the involution. It is not difficult to verify that each such curve is obtained in this way. Indeed, let C be hyperelliptic of genus g with double cover f : C → ℙ^{1}. Then f_{∗}O_{C} = O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−g − 1), which gives a natural imbedding

j : C ↪ ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−g − 1)) = 𝔽_{g+1}.

C is disjoint from the negative section of 𝔽_{g+1}, so we may regard C as a curve in S_{2} = ℙ(g + 1, 1, 1). The ruling of 𝔽_{g+1} cuts out the g_{2}^{1} on C, so any conjugate points p_{1} and p_{2} lie on some ruling of S_{2}.

6.2.2 A_{2g} singularities

See figure 7 for a schematic representing this case. Our general analysis implies that (C_{2}, p_{1}) is a hyperelliptic curve of genus g, where p_{1} is a fixed point of the hyperelliptic involution. Again, every such curve arises in this way. We imbed C into 𝔽_{g+1} as before. Each fixed point p_{1} of the hyperelliptic involution corresponds to a ruling tangent R to C. Blow up 𝔽_{g+1} twice to

Figure 8: Some local stable reductions for y^{4} = x^{4}

eliminate this tangency and let E_{1} and E_{2} denote the corresponding exceptional divisors. Then blow down all the curves with self-intersection < −1, i.e. the proper transforms of the negative section, R, and E_{1}. We obtain the weighted projective space ℙ(2g + 1, 2, 1) and E_{2} corresponds to B_{2}.

6.2.3 y^{4} = x^{4}

See figure 8 for a schematic representing this case. Let C_{0} be the singularity y^{4} = x^{4}. Then C_{2} is a plane quartic curve and B_{2} ∩ C_{2} = {p_{1}, p_{2}, p_{3}, p_{4}} is a hyperplane section. There is a further compatibility condition to be satisfied. The branches of our singularity determine four distinct tangent directions, i.e. an element of M_{0,4}, which coincides with the element determined by (B_{2}, p_{1}, p_{2}, p_{3}, p_{4}).

Figure 9: Some local stable reductions for D_{2k+3} singularities

6.3 Singularities of quasitoric type

Applying Theorem 6.1 and emulating Theorems 6.2 and 6.3, we can describe tails for certain singularities of quasitoric type. Again, the surface tail S_{2} will be a weighted projective space of type ℙ(p, q, 1), attached to S_{1} along the same curve B_{2}. The only difference is that the tails C_{2} are no longer Cartier on S_{2}. They pass through one (or both) of the singularities of S_{2} and are smooth at these points. If necessary, we impose compatibility conditions like those of Theorem 6.3.

We illustrate this with the remaining ADE singularities. Note that the singularities A_{n}, D_{4}, E_{6}, and E_{8} are of toric type.

6.3.1 D_{2k+3} singularities

See figure 9 for a schematic representing this case. Let C_{0} be the singularity x(y^{2} −x^{2k+1}), i.e. the singularity D_{2k+3}. The surface tail S_{2} = ℙ(2, 2k +1, 1). The curve C_{2} is a Weil divisor with class L + 2B_{2} = 4(k + 1)B_{2} and contains the index-(2k + 1) singularity p_{2} ∈ S_{2}. It is hyperelliptic of genus k + 1 and C_{2} ∩ B_{2} = {p_{1}, p_{2}}, where p_{1} is a Weierstrass point and p_{2} is generic.

6.3.2 D_{2k+4} singularities, k > 0

See figure 10 for a schematic representing this case. Now let C_{0} be the singularity x(y^{2} − x^{2k+2}) for k > 0, i.e. the singularity D_{2k+4}. The surface tail S_{2} = ℙ(k + 1, 1, 1). The curve C_{2} is a Weil divisor with class 2L + B_{2} = (2k + 3)B_{2} and contains the index-(k + 1) singularity on p_{3} ∈ S_{2}. It is Figure 10: Some local stable reductions for D_{2k+4} singularities, k > 0

Figure 11: Some local stable reductions for E_{7}

hyperelliptic of genus k + 1 and C_{2} ∩ B_{2} = {p_{1}, p_{2}, p_{3}}, where p_{1} and p_{2} are conjugate under the hyperelliptic involution and p_{3} is generic.

6.3.3 E_{7}

See figure 11 for a schematic representing this case. Let C_{0} be the singularity x(x^{2}−y^{3}) = 0, i.e. the singularity of type E_{7}. The surface tail S_{2} = ℙ(2, 3, 1). The curve C_{2} is a Weil divisor with class L + 3B_{2} = 9B_{2} and contains the index-two singularity p_{2} ∈ S_{2}. It is a non-hyperelliptic genus three curve and C_{2} ∩ B_{2} = {p_{1}, p_{2}}, where 3p_{1} + p_{2} = K_{C_{2}}. In other words, p_{1} corresponds to a flex on the canonical image of C_{2}.

7 Equisingularity and boundary divisors

7.1 Blowing up equisingular deformations

Using Theorem 2.11 we compute the codimension of the locus of singularities of type (p, q):

Corollary 7.1 Assume that p ≥ q and b = gcd(p, q). Then the locus of singularities of type (p, q) has codimension

c(p, q) = \frac{1}{2}(pq + p + q − b) − ⌈\frac{p}{q}⌉ − 1.

The versal deformation space of the singularity x^{p} = y^{q} has dimension pq − p − q + 1. It remains to find a formula for the cardinality N of the set

{(i, j) : qi + pj ≥ pq, 0 ≤ j ≤ q − 2, 0 ≤ i ≤ p − 2} =
{(i, j) : si + rj ≥ brs, 0 ≤ j ≤ bs − 2, 0 ≤ i ≤ br − 2}

where p = br and q = bs. We have that

N = ∑_{j=1}^{bs−2}(⌊\frac{jr}{s}⌋ − 1)

= −(bs − 2) − ⌊\frac{(bs − 1)r}{s}⌋ + ∑_{j=0}^{bs−1}⌊\frac{jr}{s}⌋

= −bs − br + 2 + ⌈\frac{r}{s}⌉ + \frac{b}{2}(brs − r − s + 1)
= 2 + ⌈\frac{r}{s}⌉ + \frac{b}{2}(brs − 3r − 3s + 1)

In the third step, we applied the summation formula

∑_{j=0}^{bs−1}⌊\frac{jr}{s}⌋ = \frac{b}{2}(brs − r − s + 1).

Therefore, the codimension equals

\frac{1}{2}(b^{2}rs + br + bs − b) − ⌈\frac{r}{s}⌉ − 1

which agrees with the formula. □

Now we can state the main result of this section.

Theorem 7.2 Let C_{0} be a singularity of type (p, q) and let Z ⊂ T_{C_{0}} denote the tails described in Theorems 6.2 and 6.3. Then

dim(Z) = c(p, q) − 1

where c(p, q) is the codimension of the locus of curves of type (p, q).

The proof of the theorem is a parameter count: for each C_{0} of type (p, q), the stable limits parametrized by Z depend on c(p, q) − 1 parameters. It suffices to compute the dimension of the corresponding linear series on ℙ(p, q, 1), minus the dimension of the automorphism group and the constraints imposed by compatibility conditions. Indeed, any deformation of the imbedding C_{2} ↪ S_{2} is induced by an automorphism of S_{2}. This follows from the fact that H^{1}(T_{S_{2}}(−C_{2})) = 0, which can be proved using Serre duality and the cohomology computations of [7] 2.3.2.

We first consider the case where p = q = b. In this case, C_{2} ⊂ ℙ^{2} is a plane curve of degree b, with a distinguished hyperplane section B_{2}. Furthermore, the compatibility condition C_{2} ∩ B_{2} ≅ C_{1} ∩ B_{1} must be satisfied, which imposes b − 3 conditions. Finally, we have to account for the automorphism group of ℙ^{2}. We obtain

dim(Z) = \frac{1}{2}(b^{2} + 3b) + 2 − (b − 3) − 8 = \frac{1}{2}(b^{2} + b − 6)

which equals c(b, b) − 1.

In the case where p = rq with r > 1, we have that S_{2} = 𝔽_{r}, C_{2} ∈ |qξ_{r}| where ξ_{r} is a positive section, and B_{2} is a distinguished ruling. The compatibility conditions are that B_{2} ∩ C_{2} = B_{1} ∩ C_{1} and the singularities of S_{1} and S_{2} coincide; this imposes q − 2 conditions. Using the dimension counts for weighted projective spaces (§ 5.1)

dim(Z) = \frac{q}{2}(rq + r + 1 + 1) + 1 − (q − 2) − (r + 5) = \frac{1}{2}(rq^{2} + rq) − r − 2

which equals c(rq, q) − 1.

Now we consider the case where q does not divide p, and set b = gcd(p, q), p = br, and q = bs. We have S_{2} = ℙ(r, s, 1) and C_{2} ∈ |brsB_{2}|. The compatibility conditions are that B_{2} ∩ C_{2} = B_{1} ∩ C_{1} and the singularities of S_{1} and S_{2} coincide; since S_{1} and S_{2} have two singularities, this imposes b − 1 conditions. Again using the dimension counts for weighted projective spaces

dim(Z) = \frac{b}{2}(brs + r + s + 1) − (b − 1) − ⌊\frac{r}{s}⌋ − 4
= \frac{1}{2}(b^{2}rs + br + bs − b) − ⌈\frac{r}{s}⌉ − 2

which equals c(p, q) − 1. □

Theorem 7.2 is consistent with the following geometric picture of the ‘map to moduli.’ Let C_{0} be an isolated plane curve singularity. Consider the rational map μ : Ver_{C_{0}} −−→ \overline{M} associating to each curve its corresponding stable limit. Of course, μ is well-defined only after choosing a linear series of projective curves representing Ver_{C_{0}} (cf. proof of Proposition 3.2). μ has indeterminacy along the locus where the curves have singularities worse than nodes. For each (p, q), the total transform of the curves of type (p, q) under μ has dimension dim(Ver_{C_{0}}) − 1, i.e. it forms a divisor in the image of μ. Note that this immediately implies the following:

Proposition 7.3 Let C_{0} be a general singularity of type (p, q). Then the closure of Z is an irreducible component of T_{C_{0}}.

Here ‘general’ means contained in an open subset of the parameter space for singularities of type (p, q); this parameter space is discussed at the end of § 2.

For a concrete linear series Λ parametrizing curves of genus g, the moduli map μ : Λ −−→ \overline{M}_{g} does not necessarily map the curves with a singularitiy of type (p, q) to a divisor in the image. For example, if Λ = |O_{ℙ^{2}}(+4)| then μ sends the tacnodal locus to a codimension-two subvariety of \overline{M}_{3} [13].

7.2 Application to boundary divisors

In this section, we apply these results to analyze boundary components for certain moduli spaces of stable log surfaces. See [18],[16], and [1] for more information on the construction of these spaces. Consider pairs (ℙ^{2}, C) where C is a smooth plane curve of degree d ≥ 4; let P_{d} denote the isomorphism classes of such pairs. Let M be the connected moduli space of smoothable stable log surfaces containing P_{d} as an open subset, and let \overline{P}_{d} denote the closure of P_{d} is M. This space is discussed in more detail in [13]. Our first result is Proposition 7.4 Assume there exists a plane curve D_{0} of degree d with a single singularity, of type (p, q), such that

1. the plane curves of degree d map surjectively onto the versal deformation space of D_{0};

2. K_{S_{1}} + B_{1} + D_{1} is ample.

Then the stable limits of the curves of type (p, q) yield a boundary divisor of \overline{P}_{d}.

By the first assumption, when we apply local stable reduction to degree d plane curves, we obtain all the tails described in Theorems 6.2 and 6.3. It also implies that the curves with a singularity of type (p, q) have the expected codimension c(p, q). The second assumption means that local and global stable reductions coincide (see the end of § 3). Theorem 7.2 implies that the corresponding stable limits yield a divisor in \overline{P}_{d}. □

For fixed (p, q), the assumptions of the proposition are satisfied for sufficiently large d. As a consequence, we obtain the following theorem:

Theorem 7.5 Fix (p, q). For d sufficiently large, the stable reductions of curves with a singularity of type (p, q) yield a boundary divisor of \overline{P}_{d}.

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Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637 USA email address: hassett@math.uchicago.edu