JAROD ALPER, MAKSYM FEDORCHUK, AND DAVID ISHII SMYTH

Transcription

1 SINGULARITIES WITH G m -ACTION AND THE LOG MINIMAL MODEL PROGRAM FOR M g JAROD ALPER, MAKSYM FEDORCHUK, AND DAVID ISHII SMYTH Abstract. We give a precise formulation of the modularity principle for the log canonical models M g(α) := Proj d 0 H0 (M g, d(k Mg + αδ) ) of the moduli space of stable curves. We define a new invariant of Gorenstein curve singularities with G m-action which can be used to predict the critical α-value at which a singularity first arises in the modular interpretation of M g(α). We compute these critical α-values for large classes of singularities with G m-action, including all ADE, toric planar, and unibranch Gorenstein singularities, and use these results to give a conjectural outline of the log MMP for M g. Contents 1. Introduction 1. Curve singularities with G m -action 8 3. Character calculations Connections to Intersection Theory 9 5. Connections to Geometric Invariant Theory 3 References Introduction In [Has05], Hassett and Keel initiated a program to give modular interpretations to the log canonical models (1) M g (α) := Proj d 0 H 0 (M g, d(k Mg + αδ) ), for all α [0, 1] Q such that K Mg + αδ is effective. Subsequent contributions to the Hassett- Keel program have established the modularity of various log canonical models by realizing them as GIT quotients of Hilbert schemes of pluricanonically embedded curves [HH09, HH13, HL10]. These works have established the modularity of the log canonical models M g (α) for all α when g =, 3, and for α 7/10 ɛ when g 4. For additional partial results when g = 4, 5, 6, see [Fed1, CMJL1a, CMJL1b, FS13b, Mül13]. Unfortunately, the GIT approach is limited by the absence of natural GIT quotient descriptions of M g (α) for low values of α, as well as by the increasing difficulty of the GIT stability analysis even in cases where quotient descriptions do exist. It is therefore natural to seek an abstract formulation of the goal of the Hassett-Keel program. The first author was partially supported by an NSF Postdoctoral Research Fellowship under Grant No The second author was partially supported by NSF grant DMS The third author was partially supported by NSF grant DMS

2 ALPER, FEDORCHUK, AND SMYTH 1.1. Modularity Principle for M g (α). Informally, the Hassett-Keel program suggests the existence of stacks of singular curves whose moduli spaces are the log canonical models M g (α). In order to make this precise, we must introduce some terminology. First, we let U g denote the stack of complete Gorenstein 1 curves of arithmetic genus g with ample dualizing line bundle. We let π : C g U g denote the universal curve and consider the following line bundles on U g : () λ m := c 1 (π ω m π ), for m 1, K := λ 13λ 1, δ := 13λ 1 λ. Remark 1.1. A Grothendieck-Riemann-Roch computation shows that the restriction of K and δ to M g is the usual canonical class and boundary divisor class, respectively; see [HM8, Section ] and [HM98, Section 3.E] for details of this computation. Recall from [Alp08] that a morphism φ : X X from an algebraic stack to an algebraic space is a good moduli space if φ O X = O X and φ : QCoh(X ) QCoh(X) is an exact functor. The formal properties of good moduli spaces will not play a significant role in this paper, but we encourage the reader to keep in mind that the canonical example of a good moduli space is the morphism [X ss /G] X ss / G from a GIT quotient stack to the GIT quotient. Finally, since M g (α) is defined as the Proj of the graded section ring of the Q-line bundle K Mg +αδ on M g, there exists a natural Q-line bundle O Mg(α) (1) on M g(α) such that f (K Mg + αδ) = O Mg(α) (1) under the rational map f : M g M g (α). With all this notation, we may consider the goal of the Hassett-Keel program as the verification of the following modularity principle. Principle 1. (Modularity principle for the log MMP for M g ). For α [0, 1] Q such that K Mg + αδ is effective, there exists an open normal substack M g (α) U g and a morphism φ : M g (α) M g (α) satisfying: (1) φ is a good moduli space. () φ O Mg(α) (1) M g(α) M g ( K Mg + αδ ) Mg(α) M g. To verify the modularity principle in this abstract setting, one would like a natural recipe for defining the open substacks M g (α) U g. One step in this recipe should be a method for predicting which singular curves arise at a given value of α. In this paper, we construct a new invariant of curve singularities with G m -action, namely the α-invariant, which is tailored to answer this question. We compute the α-invariant for several natural classes of singularities, and use our computations to give a conjectural outline of the log MMP down to α = 5/9. In fact, these computations have already been deployed as a useful guiding heuristic in recent works on the log minimal model program [CMJL1b, AFSvdW13]. 1.. Curves with G m -action in the log MMP for M g. Before defining the α-invariant, we should explain why singularities with G m -action play such a special role in our analysis. In fact, there is good reason to expect that all reduced curve singularities arising in the Hassett-Keel 1 Our decision to focus on Gorenstein curves is motivated in part by experience (we have yet to see a non- Gorenstein curve appear in the Hassett-Keel program) and in part by technical convenience (the Hodge line bundles which play a central role in this paper are naturally defined over the locus of Gorenstein curves).

3 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 3 program admit a G m -action. Indeed, at each threshold value of α, we expect a diagram M g (α +ɛ) i + M g (α) i M g (α -ɛ) M g (α +ɛ) j + M g (α) j M g (α -ɛ) where i +, i are open immersions, the vertical maps are good moduli spaces, and j +, j are projective morphisms induced by i +, i. Crucially, the closed points of M g (α) ( M g (α +ɛ) M g (α -ɛ) ) must all have infinite stabilizers, i.e. must correspond to curves admitting a G m -action. Indeed, the only way an open immersion of stacks can induce a projective morphism of good moduli spaces is for all new points of the larger stack to be identified with certain points of the smaller stack, and this requires the new closed points of the larger stack to have infinite stabilizers. There are two reasons for believing that each transition of the Hassett-Keel program should entail a diagram of this kind. First, we believe the log MMP for M g should be modeled after a variation of GIT problem, and VGIT problems always give rise to such diagrams [DH98, Tha96]. Indeed, as one varies the polarization in VGIT, the semistable locus jumps at certain walls, and the corresponding open inclusions of the semistable loci induce projective morphisms of the corresponding quotients. Second, the work of Hassett-Hyeon and Hyeon-Morrison produces precisely such a picture at the first two critical thresholds. Indeed, at α = 9/11, the points of M g (α) ( M g (α +ɛ) M g (α -ɛ) ) correspond to curves with rational cuspidal tails (Figure 1) which contain all elliptic tails and cuspidal curves in their basins of attraction [HM10]; see [HH13, Definition 5.] for the definition of the basins of attraction. At α = 7/10, the points of M g (α) ( M g (α +ɛ) M g (α -ɛ) ) correspond to curves with rational tacnodal bridges, and contain all elliptic bridges and tacnodal curves in their basins of attraction [HH13]. g=1 A A g=0 Figure 1. Rational cuspidal tail is an isotrivial degeneration of both elliptic tails and cuspidal curves. Since these curves correspond to closed points of a stack with a good moduli space, their stabilizers must be linearly reductive. But if C is any reduced curve with positive dimensional linearly reductive automorphism group Aut(C), the connected component of the identity of Aut(C) is a non-trivial torus, i.e. C admits a G m-action.

4 4 ALPER, FEDORCHUK, AND SMYTH 1.3. α-invariants of curves with G m -action. In moduli theory, the presence of objects with infinite automorphism groups is usually considered bad news, but the key idea of this paper is to make a virtue of necessity by exploiting this feature of the log MMP to obtain a priori information on the definition of the stacks M g (α). To see how this works, recall that if [C] U g is any point and L is a line bundle defined in a neighborhood of [C], then the natural action of Aut(C) on the fiber L [C] induces a character Aut(C) G m. If η : G m Aut(C) is any one-parameter subgroup, then there is an induced character G m G m which is necessarily of the form z z n for some integer n Z. For a given curve C, a non-trivial one-parameter subgroup η : G m Aut(C), and a line bundle L Pic(U g ), we denote this integer by χ L (C, η). If Aut(C) G m, we write simply χ L (C), where η : G m Aut(C) is understood to be the identity. Note that χ L (C) is only defined up to sign due to the choice of isomorphism Aut(C) G m, but the ratios χ L (C)/χ M (C), for any two line bundles L and M, are welldefined and this is all we ultimately use. If L = λ m, we write simply χ m (C, η) or χ m (C) if η is understood. These characters are connected to the modularity principle by the following observation. Proposition 1.3 (Key Observation). Assume the modularity principle holds. Suppose in addition that the locus of worse-than-nodal curves in M g (α) has codimension at least. Then for any [C] M g (α) and any one-parameter subgroup η : G m Aut(C), we have Proof. If the modularity principle holds, then χ (K+αδ) (C, η) = 0. φ O Mg(α) (1) M g(α) M g ( K Mg + αδ ) Mg(α) M g, where φ : M g (α) M g (α) is a good moduli space. On the other hand, since the codimension of M g (α) \ ( ) M g (α) M g in Mg (α) is at least and M g (α) is normal, the line bundle K +αδ = ( α)λ +(13α 13)λ 1 is the unique locally-free extension of (K Mg +αδ) Mg(α) M g to M g (α). Here we use the fact that the Hodge line bundles λ m are defined over all of U g and hence over M g (α). It follows that K +αδ φ O (1). However, for any point [C] M Mg(α) g(α), the induced action of Aut(C) on the fiber of φ O Mg(α) (1) is necessarily trivial. This finishes the proof. What makes this observation useful is the fact that the character χ (K+αδ) (C, η) can be computed in terms of the intrinsic geometry of C, without knowing the definition of M g (α) a priori. Indeed, by definition χ (K+αδ) (C, η) = ( α)χ (C, η) + (13α 13)χ 1 (C, η). Now the characters on the right are manifestly computable in terms of the intrinsic geometry of (C, η), indeed χ m (C, η) is simply the determinant of the action of η on H 0 (C, ωc m ). Using this, we may define the α-invariant of a curve with G m -action to be the unique rational number α(c, η) such that this character is zero. Definition 1.4 (α-invariant). The α-invariant of a curve C U g with G m -action η : G m Aut(C) is the rational number ( ) χ (C, η) 13 χ 1 (C, η) α(c, η) := ( ). χ (C, η) 13 χ 1 (C, η)

5 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 5 If χ 1 (C, η) = 0, the α-invariant is undefined. If Aut(C) G m, we write α(c) for the α-invariant α(c, η) where η : G m Aut(C) is understood to be the identity. We conclude that, as long as the α-invariant of a curve with G m -action is defined, it is the only α-value at which the curve can show up in M g (α). Corollary 1.5. Assume the modularity principle holds and let [C] M g (α) be any point. If C admits a G m -action η such that χ 1 (C, η) 0, then α = α(c, η). Proof. By Proposition 1.3, we must have χ (K+αδ) (C, η) = ( α)χ (C, η) + (13α 13)χ 1 (C, η) = 0. Solving for α, we obtain the desired statement. While the α-invariant α(c, η) can be computed for any explicitly given curve with G m -action, certain classes of curves play a distinguished role in announcing critical thresholds of the log minimal model program. In this paper, we isolate three classes of complete curves with G m - action, namely Ô-atoms, ÔS -atoms, and A i/j -atoms. These will be defined carefully in Section (Definitions.6,.9, and.1), but we give here an informal description of these curves and of their geometric significance in the context of the log minimal model program. We have already seen that M g (9/11) contains curves of the form C 0 E, where E is a rational cuspidal tail, and that M g (7/10) contains curves of the form C 0 E, where E is a rational tacnodal bridge. The Ô-atom is simply a generalization of this construction to an arbitrary Gorenstein curve singularity Ô with G m-action. Indeed, if the singularity Ô has b branches and δ-invariant δ(p), an Ô-atom is simply a curve of the form C = E 1... E b C 0, where C 0 is any smooth curve of genus g δ(p) b + 1 and E 1,..., E b are rational curves attached to C 0 nodally and meeting in a singularity analytically isomorphic to Ô (see Figure for an example of J 10 -atom). The G m -action on Ô extends to define a one-parameter subgroup η : G m Aut(C), and we define the α-invariant of the Ô-atom to be α(c, η). Using this definition, the α-invariant of a cusp and a tacnode are 9/11 and 7/10 respectively, and in general we expect the α-invariant associated to a singularity to be the first α-value at which the generic form of this singularity appears in the modular interpretation of M g (α). (We discuss non-generic forms of singularities in the next paragraph.) We compute the α-invariants of a broad range of singularities and the results are collected in Table 1 of Section 3.4. E 1 E E 3 y 3 = x 6 C 0 Figure. An atom of the y 3 = x 6 singularity (also called J 10 -atom). There are two natural generalizations of Ô-atoms which we expect to play an important role in future stages of the log minimal model program for M g. To motivate these constructions, let us consider which loci on M g must be modified at the critical value α = 19/9. The divisor is covered by curves intersecting K δ trivially, and we expect curves with genus tails to be

6 6 ALPER, FEDORCHUK, AND SMYTH replaced by dangling oscnodes, using the blow-up/blow-down procedure pictured in Figure 3. The distinguished curve with G m -action to which this dangling oscnode isotrivially specializes is the curve pictured in Figure 4. P 1 g = y = x 6 Figure 3. Given a smoothing of a curve with a genus tail attached at an arbitrary point p, after blowing up the conjugate point of p and contracting the genus curve, one obtains a dangling P 1 attached at an oscnode. A 5 : y = x 6 Figure 4. Dangling A 5 -atom. Note that this curve is not an oscnodal atom because one of the branches of the oscnode dangles, i.e. is not nodally attached to the rest of the curve. This motivates a generalization of the construction of the Ô-atom, where some subset S of the branches of Ô are allowed to dangle, and we call such curves ÔS -atoms. The α-invariant of an ÔS -atom depends on S, and this gives rise to important subtleties in the unfolding of the log minimal model program. For example, while the α-invariant of the dangling oscnodal atom is 19/9, the α-invariant of the standard oscnodal atom is 17/8, reflecting the fact that genus bridges attached at conjugate Weierstrass points are not replaced until α = 17/8. The α-invariants for a wide selection of Ô S -atoms are displayed in Table of Section 3.4. A second locus that must be modified at α = 19/9 is the locus of genus two tails attached tacnodally at a Weierstrass point. Of course, this locus does not appear in M g, but it appears in M g (7/10) and subsequent models, and one can check that it is covered by curves intersecting K δ trivially. We expect these curves to be replaced by tacnodally attached ramphoid tails. The corresponding distinguished curve with G m -action to which such curves specialize is pictured in Figure 5. This motivates consideration of A i/j -atoms constructed by concatenating

7 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 7 A i and A j -singularities along rational branches (Definition.1). The α-invariants associated to A i/j -atoms are listed in Table. A 3 A 4 Figure 5. A 3/4 -atom. In summary, using the computations of α-invariants of Ô-atoms, ÔS -atoms, and A i/j -atoms, we can assemble a fairly comprehensive set of predictions for the log minimal model program. While there are innumerable further variations on these constructions, we expect that these three varieties of singular curves with G m -action are sufficient to give a skeletal outline of the log minimal model program for α 5/ The case of pointed curves. We note that there is a simple way to generalize our results to the case of pointed curves. Namely, let U g,n denote the stack of Gorenstein curves C of arithmetic genus g with smooth marked points p 1,..., p n C such that ω C ( n i=1 p i) is ample. Let π : C g,n U g,n denote the universal curve and σ i : U g,n C g,n denote the universal sections. Then we can define ( ( n ) m λ m := c 1 π ω π σ i ), for m 1, (3) i=1 ψ i := σ i (ω π ), ψ := n ψ i, i=1 K := λ 13λ 1 ψ, δ := 13λ 1 λ + ψ. With this notation, we will be able to define and study the α-invariants of pointed curves with G m -action, and relate them to the α-invariants of unmarked curves. Outline of the paper. Let us now give a roadmap for the rest of the paper. In Section, we discuss singularities with G m -action, and define Ô-atoms, ÔS -atoms, and A i/j -atoms, and the associated α-invariants. In Section 3, we explain how to compute these α-invariants: In 3.1, we compute the α-invariants of a wide range of singularities, including ADE, toric planar, and unibranch Gorenstein singularities. In 3., we give an alternative method for computing χ K and χ δ, under additional assumptions on the deformation space of the curve. While the results of these calculations are ultimately derivable from the calculations in 3.1, they help to illuminate the geometric significance of α-invariants. We close Section 3 by summarizing our calculations of the α-invariants in Tables 1 and. We use them to generate predictions for the log MMP, and give a complete outline of the transitions of the log MMP for α 5/9 in Table 3. Finally, in Sections 4 and 5, we explore the connection between α-invariants and important invariants coming from intersection theory and geometric invariant theory. In Section 4, we explain how the α-invariant is related to slopes of families in the locus of stable limits of a singularity. In Section 5, we explain how Hilbert-Mumford indices are related to the characters. We use this connection to prove some new instability results for rational ribbons of Bayer Eisenbud. These computations were a key inspiration for the results of [AFS13].

8 8 ALPER, FEDORCHUK, AND SMYTH Notation. We work over an algebraically closed field k of characteristic 0 and denote by G m the multiplicative group of units in k. A curve is a connected finite type scheme over k of pure dimension 1. Acknowledgements. We thank Anand Deopurkar, David Hyeon, Filippo Viviani, and Fred van der Wyck for stimulating discussions. We thank Igor Dolgachev and Henry Pinkham for pointing out a number of helpful references. We thank David Hyeon for sharing an early version of his preprint [Hye10]. We also thank the referee for detailed remarks that significantly improved the exposition.. Curve singularities with G m -action In this section, we collect definitions and preliminary results concerning reduced curve singularities with G m -action. We begin by recalling the definition of the dualizing sheaf of a singular curve in Section.1. We use this description in Section. to show that the analytic isomorphism type of a Gorenstein curve singularity with G m -action uniquely determines an affine curve with G m -action. This leads to a precise definition in Section.3 of an Ô-atom for any Gorenstein singularity Ô with G m-action..1. Dualizing sheaf of a singular curve. We recall duality theory on singular curves as in [AK70, VIII], which treats reduced curves over any field. The summary of this theory can also be found in [BG80, p.44], as well as in [Ser88, Ch.IV.9] (at least for irreducible curves) and in [BHPVdV04, Prop.6.] for reduced curves on smooth surfaces. Given a reduced curve C, denote by ν : C C the normalization of C and consider the sheaf M C of rational differentials on C. Then by [AK70, VIII, Prop.1.16] the dualizing sheaf of C can be defined as the subsheaf ω C ν M C of the rational differentials ω satisfying the following condition: For every p C and every f O C,p (4) Res q (fω) = 0. q ν 1 (p) Differentials ω satisfying (4) are called Rosenlicht differentials. A singularity p C is called Gorenstein if ω C,p is a free O X,p -module of rank 1. Suppose p C is a curve singularity with b branches. Let t 1,..., t b be the uniformizers of the branches of the normalization ÕC,p. If O C,p is Gorenstein and Ann(ÕC,p/O C,p ) = (t m1 1 ) (tm ) (tm b b ) is the conductor ideal, then ω C,p is generated by a Rosenlicht differential of the form ( ) dt 1 dt dt b (5) u 1, u t m1,..., u 1 t m b t m, b b where u i is a unit on the i th branch; see [Ser88, Ch.IV,11]. For every pair of branches i j {1,..., b}, we have a Rosenlicht differential at p defined by ( (6) ω(i, j) := 0, dt i, 0 0, dt ) j, 0 ω C,p. t i t j In addition, if the branch with the uniformizer t i is singular, then dt i t i ω C,p.

9 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 9 Recall from [Ste96, Def..1, p.385] that a curve singularity p C is decomposable if C is the union of two curves C 1 and C which lie in smooth spaces intersecting each other transversely in p. Algebraically, this means that (7) Ô C,p k[[x 1,..., x n, y 1,..., y m ]]/(I, y 1,..., y m ) (J, x 1,..., x n ), where I (x 1,..., x n ) k[x 1,..., x n ], and J (y 1,..., y m ) k[y 1,..., y m ] are the ideals of C 1 and C. Proposition.1. Every Gorenstein curve singularity, with the exception of the ordinary node xy = 0, is indecomposable. Proof. This follows from a more general result of [Ste96, p.385], but for the reader s convenience we include a self-contained proof. Suppose ÔC,p is a decomposable Gorenstein curve singularity as in (7). Let k[[t 1 ]] k[[t b ]] be the normalization of Ô C,p. By the decomposability assumption, there are two branches, say those with uniformizers t 1 and t, such that (x 1,..., x n ) 0 k[[t 1 ]] and (y 1,..., y m ) 0 k[[t ]]. Recall the definition of ω(i, j) from (6). Note that ω(1, ) ω C,p, but ω(1, ) / (x 1,..., x n )ω C,p + (y 1,..., y m )ω C,p because (dt 1 /t 1, 0 ) / ω C,p. We conclude that ω C,p must be generated by ω(1, ). If b 3, then ω(1, 3) is not a multiple of ω(1, ) and we arrive at a contradiction. If b =, then the fact that dt i /t i (for i = 1, ) is not a multiple of ω(1, ), hence not in ω C,p, implies that both branches of p are smooth. Since they intersect transversally, p X is a node... Curve singularities with G m -action. Observe that if C U g is a curve with G m -action and p C is a singular point, then there is an open affine G m -invariant subcurve X C such that p X. 3 For this reason, we focus our attention on singular affine curves with G m -action. Given an affine curve X with a distinguished singular point p X and G m -action on X fixing p, we can choose coordinates so that: (1) X Spec k[x 1,..., x n ]/I where p corresponds to the maximal ideal (x 1,..., x n ), and () G m = Spec k[λ, λ 1 ] acts by x i λ di x i and I is invariant under this action. The completion ÔX,p is called a singularity with G m -action [OW71, Pin74]. The G m -action is good if the weights d i are either all negative or all positive and gcd(d 1,..., d n ) = 1. In this situation, the completion ÔX,p is called a quasi-homogeneous singularity. Note that if the G m -action on (X, p) is good, then G m acts freely on X p. Our first goal is to show that the pair (X, p) is uniquely determined from the isomorphism type of ÔX,p, at least when ÔX,p is Gorenstein. Evidently, if (X, p) is an affine curve with good G m -action, then the induced action on the normalization X is non-trivial on every branch and hence X = Spec k[t 1 ] k[t b ], where b is the number of the branches of p X. In particular, we can describe Γ(X, O X ) as a G m - invariant subring of k[t 1 ] k[t b ]. The following proposition says that a G m -action on an indecomposable curve singularity must be good. Lemma.. Suppose (X, p) is an indecomposable singular affine curve with faithful G m -action. Then the G m -action is good. Proof. Suppose X Spec k[x 1,..., x n, y 1,..., y m ]/I is an affine reduced curve with G m -action fixing the origin such that G m acts with positive weights on (x 1,..., x n ), and with non-positive weights on (y 1,..., y m ). The G m -action lifts to the normalization of X at p. Let t 1,..., t k be 3 As the example of a nodal plane cubic illustrates, the condition that ωc is ample cannot be omitted.

10 10 ALPER, FEDORCHUK, AND SMYTH the uniformizers of the branches on which G m acts with positive weight, and let t k+1,..., t b be the uniformizers of the remaining branches. Then (x 1,..., x n ) maps to 0 in k[t k+1 ] k[t b ] and (y 1,..., y m ) maps to 0 in k[t 1 ] k[t k ]. It follows that x i y j = 0 on the normalization, for all i and j. Hence (x 1,..., x n )(y 1,..., y m ) I, which in turn implies that X is decomposable. A contradiction. Corollary.3. A faithful G m -action on a Gorenstein non-nodal singularity is necessarily good. Greuel, Martin, and Pfister gave a numerical characterization of quasi-homogeneous Gorenstein curve singularities, generalizing a quasi-homogeneity criterion for hypersurface singularities due to Saito [Sai71]. Namely, a Gorenstein curve singularity is quasi-homogeneous if and only if the Milnor number equals to the Deligne number [GMP85, Theorem.1], both of which are effectively computable [GMP85, Eqns.(1.1),(1.4)]. In particular, one can determine whether a Gorenstein curve singularity Ô is the completion of O X,p for some affine curve (X, p) with good G m -action from numerical invariants of Ô itself. Our next result shows that in fact a quasi-homogeneous Gorenstein curve singularity Ô comes from a unique affine curve (X, p) with G m -action. Proposition.4. Suppose (X, p) and (Y, q) are affine curves with faithful G m -actions such that ÔX,p and ÔY,q are isomorphic Gorenstein complete local rings, which are not nodes. Then there is a G m -equivariant isomorphism (X, p) (Y, q) of pointed affine schemes. Proof. Note that G m -actions on (X, p) and (Y, q) are good by Corollary.3. Let X = Spec k[t 1 ] k[t b ] and Ỹ = Spec k[s 1] k[s b ] be the G m -equivariant normalizations of (X, p) and (Y, q), respectively. Then the G m -actions are given by λ t i = λ αi t i and λ s i = λ βi s i, with α i, β i > 0. (We are using the fact that the number of branches is an analytic invariant of a singularity.) The isomorphism ÔX,p ÔY,q induces an isomorphism f : k[[t 1 ]] k[[t b ]] k[[s 1 ]] k[[s b ]] of the normalizations satisfying f(ôx,p) = ÔY,q. After renumbering the branches and scaling, we can assume that f(t i ) = s i + c i, s i + c i,3 s 3 i +, where c i,j k. First, we show that the weights (α i ) b i=1 and (β i) b i=1 are the same. Since ω X,p is a free rank one O X,p -module, we can choose a G m -semi-invariant generator of ω X,p of the form ( dt1,..., dt ) b t m1 1 t m, b b where (t m1 1 ) (tm ) (tm b b ) is the conductor of O X,p. By the G m -semi-invariance, α 1 (m 1 1) = = α b (m b 1). Since O X,p is not a node, at least one m i. Hence all m i. It follows that (α 1,..., α b ) are determined by (m 1,..., m b ), which are determined by the conductor ideal of O X,p, and hence form an analytic invariant. We conclude that (α i ) b i=1 = (β i) b i=1. For any λ G m, consider the following isomorphism f λ := λ f λ 1 : k[[t 1 ]] k[[t b ]] k[[s 1 ]] k[[s b ]]. We have f(ôx,p) = ÔY,q. Since ÔX,p and ÔY,q are G m -invariant subrings of k[[t 1 ]] k[[t b ]] and k[[s 1 ]] k[[s b ]], respectively, we conclude that f λ also satisfies f λ (ÔX,p) = ÔY,q for any λ 0.

11 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 11 Since λ acts with the same weight α i on t i and s i, we compute (8) f λ (t i ) = s i + c i, λ αi s i + c i,3 λ αi s 3 i +. Next, let f 0 : s i t i be the standard isomorphism of the normalizations. Set m = (t 0 ) (t b ) and n = (s 0 ) (s b ). Then for any N 0, f λ induces an isomorphism fλ N : k[[t 1 ]] k[[t b ]] / m N k[[s 1 ]] k[[s b ]] / n N ) satisfying fλ (ÔX,p N /ÔX,p m N = ÔY,q/ÔY,q n N for λ 0. Equation (8) then implies that by letting λ 0, we must have f N 0 (ÔX,p /ÔX,p m N ) = ÔY,q/ÔY,q n N. By letting N and taking the inverse limit, we conclude that f 0 (ÔX,p) = ÔY,q. The proposition now follows from the equalities Γ(X, O X ) = ÔX,p (k[t 1 ] k[t b ]) and Γ(Y, O Y ) = Ô Y,p (k[s 1 ] k[s b ]). Remark.5. Proposition.4 can be regarded as (a much easier) analog of the following result due independently to Dolgachev and Pinkham [Dol75, Pin77]: Suppose Spec k 0 A k is a normal surface with an isolated singularity and a G m -action given by the grading. Then the analytic isomorphism type of the singularity determines k 0 A k. In particular, A k can be read off from the minimal resolution of the singularity [Pin77, Theorem 5.1]..3. Atoms associated to curve singularities with G m -action. Suppose Ô is a Gorenstein non-nodal curve singularity with non-trivial G m -action. By Proposition.4, there exists a unique pointed affine curve (X, p) with good G m -action such that Ô ÔX,p. The affine scheme X admits a canonical compactification X, constructed as follows: Let {E i } b i=1 be the branches of X at p, where the enumeration is chosen once and for all. Since Ẽi A 1 and the G m -action on E i has a unique fixed point p, we can compactify Ẽi to P 1 by adding a single point p i at infinity. Compactifying each branch of X, we obtain a complete pointed curve ( X; {p i } b i=1 ) whose irreducible components have normalization isomorphic to P1, and such that G m acts on each E i by scaling with fixed points p and p i. We call ( X; {p i } b i=1 ) a G m-equivariant compactification of Ô. Definition.6 (Ô-atoms). Suppose Ô is a Gorenstein curve singularity with G m-action and G m -equivariant compactification ( X; {p i } b i=1 ). Let δ(p) be the δ-invariant. An Ô-atom of genus g is any curve C obtained by nodally attaching a smooth curve C 0 of genus g δ(p) b + 1 to each of the b branches of X at the points {p1,..., p b }. Let η : G m Aut(C) be the unique G m -action on the ÔX,p-atom C extending the G m - action on X. Then by the construction of C and Proposition.4, the characters χ 1 (C, η) and χ (C, η) (defined in Section 1.3) depend only on the complete local ring ÔX,p. When the singularity ÔX,p has a name, e.g. E 6 : y 3 x 4 = 0, we use that name to denote the corresponding atom, e.g. E 6 -atom. Definition.7. Suppose Ô is a Gorenstein curve singularity with G m-action. The α-invariant α(ô) is defined to be the α-invariant α(c, η) of any Ô-atom C. Remark.8. It would have been arguably more natural to define the α-invariant of Ô X,p - singularity in terms of the characters of the complete curve X. We chose the above definition

12 1 ALPER, FEDORCHUK, AND SMYTH so that the α-invariant agrees with the α-values at which A and A 3 -singularities first appear in M g (α) for g 3. There is a natural variant of the construction of the Ô-atom, which we expect to play an important role in the future stages of the program. Definition.9 (ÔS -atoms). Suppose Ô is a Gorenstein curve singularity with G m-action and G m -invariant compactification ( X; {p i } b i=1 ). Let δ(p) be the δ-invariant. For any subset S {1,..., b} of the branches of Ô we define an ÔS -atom of genus g to be any curve C S obtained by nodally attaching a smooth curve C 0 of genus g δ(p) S + 1 to the points {p i } i S c. (Here, S c := {1,..., b} \ S.) In this construction, we think of S as indexing the branches that dangle, and we will say that a curve with an Ô-singularity in which several of the branches are dangling rational components to have a dangling Ô-singularity. We may naturally define a collection of modified α-invariants associated to these dangling singularities. E 1 E 3 C 0 E Figure 6. Dangling D {1} 6 -singularity. Definition.10. The α-invariant α(ôs ) is defined to be the α-invariant α(c S, η) of any (ÔS, η)-atom (C S, η). Remark.11. An ÔS -atom C S with {p i } i S can be regarded as a point in U g, S, and we can define the corresponding characters χ ψi (C S ; {p i } i S ); see Section 1.4 for the definition of ψ i and Corollary 3.6 for the computation of these characters. S In general, the invariants α (Ô) will depend on the subset S, which reflects the fact that curves C S for different S may appear in the moduli stack M g (α) at different values of α. The relationship between the characters of Ô and ÔS -atoms are explained in Corollaries 3.3 and 3.6. In Table, we list the α-invariants for all dangling ADE singularities. Note that since branches of any A k or toric planar singularity are isomorphic, the only relevant feature of the subset S {1,..., n} is the size. For D k+1 singularities, we use the labeling 1 for the singular branch and for the smooth branch, and for D k+ -singularities, we use 1, for the tangent branches and 3 for the smooth branch with unique tangent direction. Similarly for the E 7 singularity, we use the labeling 1 for the singular branch and for the smooth branch. The final construction which we expect to play an important role in the log minimal model program is an atom made of two type A singularities. While it is possible to make this construction in a much greater generality, we will focus on the simple cases of A i+1/j+1 and A i+1/j -atoms, as these are the only configurations we expect to play a role for α 5/9.

13 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 13 Definition.1 (A i+1/j+1, A i+1/j -atoms). We say that a genus g curve C is an A i+1/j+1 - atom (resp. A i+1/j -atom) if the following two conditions are satisfied: (1) C is of the form C = C 0 E 1 E E 3 (resp. C = C 0 E 1 E ), where C 0 is a genus g i j curve, each E k is a smooth rational curve, E 1 meets C 0 at a node, E meets E 1 at an A i+1 -singularity, and E 3 meets E at an A j+1 -singularity (resp. E has an A j -singularity); () There is a G m -action on C which restricts to a non-trivial action on each E k. See Figure 7. A i+1 A i+1 A j+1 A j Figure 7. A i+1/j+1 and A i+1/j -atoms. Definition.13. The α-invariants α(a i+1/j+1 ) and α(a i+1/j ) are defined to be the α- invariants α(c, η) of an A i+1/j+1 -atom and A i+1/j -atom respectively. 3. Character calculations In this section, we compute the α-invariants of all ADE, toric planar, and monomial unibranch Gorenstein singularities. We also compute the α-invariants for the canonical rational ribbons with G m -action using results of Bayer and Eisenbud [BE95]. These computations are summarized in Tables 1 and in Section 3.4, where we use our results to give an outline of the log MMP for M g for α 5/9. In Section 3.1, we explain how to compute the α-invariant of a curve C with G m -action using Rosenlicht differentials to write down diagonalized bases for the vector spaces H 0 (C, ω m C ). Under certain hypotheses on C, there is a somewhat more geometric method of computing α-invariants. Recall from () that K and δ are formally defined by the equations K := λ 13λ 1, δ := 13λ 1 λ, and that these classes restrict to the usual canonical class and boundary divisor on M g. These functorial interpretations cannot be extended to all of U g, but they do extend under additional assumptions on the deformation space of C. In Section 3., we will see that if C has a smooth deformation space, then the fiber K [C] can be identified with det(t 1 (C)) det(g), where g is the adjoint representation of Aut(C), and we will use this identification to compute χ K (C) for A i/j -atoms. Similarly, we will see that if the locus of singular deformations of C is cut out formally locally by a single equation and generically parameterizes nodal curves, then χ δ (C) can be computed from the weight of this equation, and we use this to compute χ δ (C) for A i/j -atoms. It is clear that any two of the characters χ 1 (C), χ (C), χ K (C), χ δ (C) determine the others, and are therefore sufficient to determine the α-invariant. Roughly speaking, the characters

14 14 ALPER, FEDORCHUK, AND SMYTH χ i (C) are easier to compute for singularities described intrinsically, i.e. in terms of the subalgebra of functions that descend from the normalization. By contrast, χ K (C) and χ δ (C) are easier to compute for singularities described by extrinsic equations, from which the deformation space and discriminant can be explicitly described Computing characters χ i. In this section, we use Rosenlicht differentials to compute α- invariants for several classes of Gorenstein singularities. We begin with the ADE singularities, which we encounter in the initial stages of the Hassett-Keel program. Next, we compute the α- invariant for an arbitrary unibranch Gorenstein singularity, and then do a multi-branch example of the elliptic m-fold points, which play a prominent role in the log MMP for M 1,n [Smy11]. We begin by explaining how to algorithmically compute the characters χ i (C) in the cases when C is an ÔS -atom The setup. We keep the notation of Section.3. Suppose ÔX,p is a Gorenstein curve singularity with G m -action and G m -equivariant compactification ( X; {p i } b i=1 ). Recall that for S {1,..., b}, we define the ÔS X,p -atom as the union CS = X C 0, where X is nodally attached at p i to a point q i C 0 for i S c. Suppose X = Spec k[x 1,..., x n ]/I and X = Spec k[t 1 ] k[t b ]. Then the normalization map ν : X X is given by xj ( f 1j (t 1 ),... f bj (t b ) ), for some monomials f ij (t i ). In what follows, we also let s i = 1/t i to be the local coordinate around p i on X. The G m -action on X extends to X as η : λ t i = λ αi t i for some integers α i 1. The weight of a semi-invariant function (or differential) f with respect to this action will be called η-weight and denoted wt η (f). Note that, α i deg f ij = wt η (x j ) for all i and j. Since X is a Gorenstein curve with G m -action, we can find a G m -semi-invariant generator of ω X,p, which we denote by ω 0 (X), or simply ω 0. After rescaling t i we have ( dt1 ω 0 =, dt,..., dt ) b t m1 1 t m t m, b b where b i=1 (tmi i ) is the conductor ideal. Note that b i=1 m i = δ(p) by [Ser88, Ch.IV,11]. We make the following observation: Lemma 3.1. H 0 ( X, ω X) is spanned by G m -semi-invariant differentials of the form fω 0, where f k[x 1,..., x n ] ranges over all monomials satisfying wt η (fω 0 ) > 0. Proof. This is clear. For example, to prove the weight condition, note that if fω 0 has a local equation dt i /t a i around 0 Ẽi, then its equation around p i is s a i ds i. It follows that fω 0 is regular at p i if and only if a if and only if the η-weight of fω 0 is positive. Lemma 3.. There is a G m -invariant decomposition H 0 (C S, ω C S ) H 0 ( X, ω X) W 0, where W 0 is the space of η-weight 0 differentials and H 0 ( X, ω X) are regular differentials on X extended by zero to C 0. Proof. By definition, H 0 ( C S ) ( (, ω C S = {(w, u) H 0 X, ω X i S c p i ) ) H 0( C 0, ω C0 ( i S c q i ) ) : Res pi w +Res qi u = 0}. 4 A final point of interest: Using χk and χ δ, one can define α-invariants for non-gorenstein curves, provided they have a smooth deformation space and the locus of worse-than-nodal deformations has codimension at least. This may be of interest for later stages in the log minimal model program.

15 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 15 A global section of H 0( X, ω X( i S p i) ) is a Rosenlicht differential on X with at worst simple c poles at {p i } i S c. By Lemma 3.1, G m -semi-invariant elements in H 0 ( X, ω X) have positive η-weight. On the other hand, any semi-invariant differential in H 0( X, ω X( i S p i) ) with a c simple pole at some p i has local equation ds i /s i = dt i /t i and hence has η-weight 0. Corollary 3.3 (Computing λ 1 ). For any S {1,..., b}, the character λ 1 (C S ) is the sum of the weights of the G m -action on the vector space H 0 ( X, ω X). In other words, λ 1 (C S ) = λ 1 ( X). In the same vein, we can algorithmically compute λ (C S ). Lemma 3.4. Suppose p a ( X). Then there is a G m -invariant decomposition H 0 (C S, ω C S ) H0 ( X, ω X) U + U 0, where U 0 is the space of η-weight 0 quadratic differentials and H 0 ( X, ω X) are regular quadratic differentials on X extended by zero to C 0. Moreover, the weights of the G m -action on U + are precisely {α i } i S c. Remark 3.5. The Gorenstein singular curves X with p a ( X) = 1 are precisely elliptic m-fold points; see [Smy11]. The cases of m = 1 and m = (the cusp and the tacnode) are considered in 3.1. and 3.1.3, and the case of m 3 is considered separately in Section Proof of Lemma 3.4. Note that p a ( X) implies that m i 3 for some i {1,..., b}. Clearly, H 0 ( X, ω X) is a G m -invariant subspace of H 0 (C S, ω C S ) consisting of quadratic differentials whose local equation at 0 Ẽi is (dt i ) /t ai i for some a i 4. The quadratic differentials of weight 0 are exactly those that have local equation c(dt i ) /t i around 0 Ẽi (with c possibly zero) for each i C S. It follows that H 0 (C S, ω C )/ ( H 0 ( X, ω X) ) U S 0 is spanned by Gm -semiinvariant quadratic differentials which have local equation (dt i ) /t 3 i on some Ẽi with i S c. It remains to show that there are precisely S c linearly independent quadratic differentials of this form and to compute their weight. In doing so, we repeatedly use the differentials ω(i, j) introduced in Equation (6). If b = 1, then ω 0 = dt/t m+1, where m is the largest gap of the semigroup of vanishing orders in ÔX,p. Since m 3, t m 1 ÔX,p and we conclude that (dt) /t 3 = t m 1 ω0 ω X,p. If S =, then (dt) /t 3 spans U + and has weight α 1, as desired. Consider now the case b. First, we note that if E i is a singular branch, then dt i /t i ω X,p. It follows that u(i) := (dt i /t i )ω(i, i+1) = (dt i) /t 3 i ω X,p. We conclude that for every singular branch E i with i S c, the differential u(i) extended by zero to C S lies in H 0 (C S, ω C ). In S particular, we are done if all the branches E i with i S c are singular. Suppose that there is a smooth branch E i with i S c, but not all branches are smooth and pairwise tangent. If m i 3, take a G m -semi-invariant function f on X whose local equation along E i is exactly t i and which vanishes to order or higher along a branch E j for some j i (in particular, either α i > α j or f 0 on E j ). Then u(i) := f mi ω 0 ω(i, j) = (..., (dt i ) /t 3 i,..., c(dt j ) /(t d j ),... ), where d 4 and c is possibly zero. If m i =, take j such that m j 3 and define u(i) := ω 0 ω(i, j) = (..., (dt i ) /t 3 i,..., (dt j ) /(t mj+1 j ),... ). Clearly, the constructed differentials {u(i)} i S c H 0 (C S, ω C ) are linearly independent and S have weights {α i } i S c.

16 16 ALPER, FEDORCHUK, AND SMYTH Finally, suppose all branches of p X are smooth and pairwise tangent. Then modulo (t 1) (t b ) the algebra of regular function on X is generated by f = (t 1,..., t b ). In particular, α 1 = = α b, and so m 1 = = m b 3. One sees at once that f mi ω 0 ω(i, j) = ( 0, (dt i ) /t 3 i, 0 0, (dt j ) /t 3 j, 0 ), f( 0, dt i /t i, 0 0, dt j /t j, 0 ) = ( 0, (dt i ) /t 3 i, 0 0, (dt j ) /t 3 j, 0 ), and where i j S c, span U +. Therefore U + has dimension S c and the claim follows. Corollary 3.6 (Computing λ ). Suppose p a ( X). For any S {1,..., b}, the character λ (C S ) is the sum of the weights of the G m -action on the vector space H 0 ( X, ω X) and i S c α i. The character of the S -pointed curve (C S ; {p k } k S ) computed with respect to the cotangent line bundle ψ i is In particular, χ ψi (C S ; {p k } k S ) = α i. λ (C S ) = λ ( X) + i S c α i = λ (C) i S χ ψi (C S ; {p k } k S ). Proof. The first assertion follows directly from Lemma 3.. The second assertion follows from the observation that the fiber of ψ i at (C S ; {p k } k S ) is ds i = dt i /t i and so G m acts on it with weight α i A k -singularity. Let C = C 0 X, where X Spec k[x, y]/(y x k+1 ), be an A k+1 - atom. The G m -action on C is η : λ (x, y) = (λ x, λ (k+1) y). The normalization map is given by (x, y) (t, t k+1 ) and the G m -action extends as t λ 1 t. It is easy to see that ω 0 := dt/t k is a G m -semi-invariant generator for ω X at the cusp, of η-weight k 1. By Lemma 3.1, a basis of H 0 ( X, ω X) diagonalizing the G m -action is x i w 0 0 i k 1. The η-weights of this basis are (k 1), (k 3),..., 1. Thus by Corollary 3.3, the character χ 1 is χ 1 (A k ) = k (i 1) = k. Similarly, if k, a basis of H 0 ( X, ω X) diagonalizing the G m -action is x i ω 0, x j yω 0 0 i k, 0 j k 3. i=1 Thus, by Corollary 3.6 the character χ is given by χ (A k ) = k 3 (4k i) + (k 3 j) + 1 = 5k 4k + 1. k i=0 j=0 If k = 1, then the only weight space of H 0 (C, ω C ) with non-zero weight is spanned by (dt) /t 4, hence χ (A ) = A k+1 -singularity. Next, we consider an A k+ -atom C = C 0 X, where X Spec k[x, y]/(y x k+ ). The G m -action on C is η : λ (x, y) = (λ 1 x, λ (k+1) y). The normalization map is (x, y) ((t 1, t ), (t k+1 1, t k+1 )) and the G m -action extends as λ (t 1, t ) = (λ 1 t 1, λ 1 t ). It is easy to see that ω 0 := ( dt 1 /t k+1 1, dt /t k+1 ) is a Gm -semi-invariant generator for ω X at

17 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 17 the tacnode, of η-weight k. If k, we may write down bases of H 0 ( X, ω X) and H 0 ( X, ω X) diagonalizing the G m -action as H 0 ( X, ω X) = x i ω 0 0 i k 1, H 0 ( X, ω X) = x i w 0, x j yw 0 0 i k, 0 j k 3. It follows by Corollaries 3.3 and 3.6 that k 1 χ 1 (A k+1 ) = (k i) = k + k, χ (A k+1 ) = i=0 k 3 (k i) + (k 1 j) + (1 + 1) = 5k + k. k i=0 j=0 (N.B. If k = 1, we need to modify the computation of χ. Namely, we observe that the only weight spaces of H 0 (C, ω C ) with non-zero weights are spanned by ((dt 1) /t 4 1, (dt ) /t 4 ) and ((dt 1 ) /t 3 1, (dt ) /t 3 ). Hence χ (A 3 ) = 3.) D k+1 -singularity. Here, X Spec k[x, y]/x(y x k 1 ) and the normalization map is given by (x, y) ( (t 1, 0), (t k 1 1, t ) ) with the G m -action on the normalization being λ (t 1, t ) = (λ 1 t 1, λ (k 1) t ). A G m -semi-invariant generator for ω X at the singularity is ω 0 := ( dt 1 /t k 1, dt /t), of η-weight k 1. It follows that the bases of H 0 ( X, ω X) and H 0 ( X, ω X) diagonalizing the G m -action are H 0 ( X, ω X) = x i ω 0 0 i k 1, H 0 ( X, ω X) = x i w 0, x j yw 0 0 i k, 1 j k. Therefore, by Corollaries 3.3 and 3.6 k 1 χ 1 (D k+1 ) = (k 1 i) = k, χ (D k+1 ) = i=0 k (4k i) + (k 1 j) + (1 + k 1) = 5k k. k i=0 j= D k+ -singularity. Consider a D k+ -atom C = C 0 X, where we take X k[x, y]/x(y x k ). The normalization map is given by (x, y) ( (t 1, t, 0), (t k 1, t k, t 3 ) ). The G m -action is given by λ (t 1, t, t 3 ) = (λ 1 t 1, λ 1 t, λ k t 3 ) and the G m -semi-invariant generator of ω X at the singularity is ω 0 := ( dt 1 /t k+1 1, dt /t k+1, dt 3 /t3), of η-weight k. The bases of H 0 ( X, ω X) and H 0 ( X, ω X) diagonalizing the G m -action are H 0 ( X, ω X) = x i ω 0 0 i k 1, H 0 ( X, ω X) = x i w 0, x j yw 0 0 i k, 1 j k. Therefore, by Corollaries 3.3 and 3.6 k 1 χ 1 (D k+ ) = (k i) = k + k, χ (D k+ ) = i=0 k (k i) + (k j) + ( k) = 5k + 3k. k i=0 j=1

18 18 ALPER, FEDORCHUK, AND SMYTH Exceptional simple singularities: E 6, E 7, E 8. To complete a list of the χ i characters of all ADE singularities, it remains to consider the cases of E 6, E 7, E 8 singularities. Among these, E 6 : y 3 x 4 = 0 and E 8 : y 3 x 5 = 0 are unibranch. We compute the characters of all unibranch curve singularities with G m -action in Section 3.1.7, but we list here the characters of E 6 and E 8 -atoms for the reader s convenience: χ 1 (E 6 ) = 8, χ (E 6 ) = 33. χ 1 (E 8 ) = 14, χ (E 8 ) = 63. Let C = C 0 X be an E 7 -atom, where X Spec k[x, y]/y(y x 3 ). The normalization map is given by (x, y) ( (t 1, t ), (t 3 1, 0) ) and the G m -action by λ (t 1, t ) = (λ 1 t 1, λ t ). The generator of ω X at the singularity is ω 0 := ( dt 1 /t 5 1, dt /t 3 ), of η-weight 4. The bases of H 0 ( X, ω X) and H 0 ( X, ω X) diagonalizing the G m -action are It follows that H 0 ( X, ω X) = ω 0, xω 0, yω 0, H 0 ( X, ω X) = ω 0, xω 0, x ω 0, yω 0, xyω 0, y ω 0 χ 1 (E 7 ) = = 7, χ (E 7 ) = ( ) = Monomial unibranch singularities. Suppose Γ = Z 0 {b 1,..., b g } is a semigroup containing 0. Let X be the G m -equivariant compactification of X Spec k[t n : n Γ], with the G m -action given by η : λ t = λ 1 t. It is easy to check that p a ( X) = g, the normalization of X is P 1, and X has an isolated monomial unibranch singularity at t = 0 with the gap sequence {b 1,..., b g }. From now on we assume that X is Gorenstein, which by [Kun70] is equivalent to Γ being a symmetric semigroup: n {b 1,..., b g } g 1 n / {b 1,..., b g }. In particular, b g = g 1. Evidently, a G m -semi-invariant generator for ω X in a neighborhood of zero is given by dt/t bg+1. Therefore, we can write down the bases of H 0 ( X, ω X) and H 0 ( X, ω X) diagonalizing the G m -actions as H 0 ( X, dt ω X) = t b1+1, dt t b+1,..., dt t bg+1, H 0 ( X, ω X) (dt) = t bg+ j : j {0,..., b g } {b 1,..., b g }. From this, we compute (9) (10) χ 1 ( X) = χ ( X) = g b i, i=1 b g (b g j) j=0 g (b g b i ) = (g 1) + i=1 g b i 1. i=1

19 SINGULARITIES WITH G m-action AND THE LOG MMP FOR M g 19 If C = C 0 X is an ÔX,p-atom, then by Corollaries 3.3 and 3.6, we have g χ 1 (C) = b i, (11) i=1 χ (C) = (g 1) + g b i. Set R(b 1,..., b g ) := (g 1) / ( g i=1 b i). Then by Definition 1.4, the α-invariant of C is i=1 α(c, η) = 11 R(b 1,..., b g ) 1 R(b 1,..., b g ). From the point of view of the log MMP for M g, it is an interesting problem to determine possible values of the α-invariants of unibranch Gorenstein singularities. Remark 3.7. In a symmetric gap sequence, one necessarily has b i i 1, so that g i=1 b i g, with the equality achieved only for the A g+1 -singularity. A similarly elementary argument shows that the only symmetric gap sequences with g i=1 b i > g g, or equivalently α(ôx,p) > 3/8, are precisely the unibranch planar singularities of types A and E, and the unique exception y 3 = x 7. In 3.1.8, we compute a closed form formula for the α-invariants of unibranch monomial singularities of embedding dimension. In particular, it is evident from this calculation that there are Gorenstein unibranch singularities with negative α-invariant. This raises a question 5 of what unibranch Gorenstein singularities have non-negative α-invariant (i.e. are expected to arise in the Hassett-Keel program). Since we do not know the answer, we state this question as an open problem: Problem 3.8. Classify (symmetric) numerical semigroups with gap sequences {b 1,..., b g } such that (g 1) R(b 1,..., b g ) = ( g i=1 b i) 11. We also note that one can associate a sub-semigroup of Z b 0 to an arbitrary curve singularity with b branches and Gorenstein singularities are characterized as those whose semigroups are symmetric [DdlM88]. It would be interesting to understand how the α-invariant of an arbitrary Gorenstein singularity can be computed in terms of its semigroup Unibranch planar singularities. For embedding dimension, the gap sequence of a monomial unibranch singularity and hence the α-invariant is easily computed. Such singularities are defined by x p = y q, or as X = Spec k[t pi+qj : i, j Z 0 ], where p and q are coprime. The gap sequence {b 1,..., b g } is the set of positive integers that cannot be expressed as pi + qj with i, j 0. The study of this sequence, e.g. finding its cardinality and the largest element is classically known in elementary number theory as the Frobenius problem [RA05]. It is well-known that the largest gap is b g = pq p q. It is also easy to see that the gap sequence is symmetric: n is a gap if and only if pq p q n is not a gap. It follows that the genus of the singularity x p = y q is g = (p 1)(q 1)/. By [BS93] (see also [Rød94]), the sum of gaps is g (1) b i = (p 1)(q 1)(pq p q 1)/1. n=1 5 The same question can be posed for arbitrary monomial unibranch singularities.