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Automorphism Groups for Quasi-homogeneous Gorenstein Surface Singularities
202, 36]43 Ž1998. JA977294
JOURNAL OF ALGEBRA ARTICLE NO.
Automorphism Groups for Quasi-homogeneous Gorenstein Surface Singularities Frieda M. Ganter U Department of Mathematics, CSV-Fresno, Fresno, California 93740 Communicated by D. A. Buchsbaum Received May 30, 1996
Let Ž X, 0. be a non]log-canonical, quasi-homogeneous surface singularity germ. And let G ; AutŽ X, 0. be the maximal reductive subgroup. In this paper we bound the order of GrCU by yP ? P, a purely topological invariant. Hurwitz’s theorem comes out as a corollary. We use the standard approach of taking the quotient space of Ž X, 0. by the action of a finite group of automorphisms acting freely on X y 04. Q 1998 Academic Press
Contents. 1. Introduction 2. Notation 3. Quasi-homogeneous singularities 4. Homogeneous singularities 5. Examples
1. INTRODUCTION Hurwitz’s theorem for a smooth projective curve of genus g G 2 bounds the order of the automorphism group,
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0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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In the local situation, if Ž X, 0. is a germ of a quasi-homogeneous Gorenstein surface singularity, with graded coordinate ring R, then we bound the order of GrCU , where G ; AutŽ R . is the maximal reductive subgroup ŽSection 3.. The bound we find, which is essentially yP ? P, is a topological invariant of Ž X, 0.. This is the content of the following theorem: THEOREM ŽTheorem 2.. Let Ž X, 0. be a non]log-canonical, quasihomogeneous Gorenstein surface singularity. Suppose there is a fixed nowhere zero holomorphic 2-form on X y 04 of degree d. Then < GrCU < F Ž42rd .ŽyP ? P .. Note that the Gorenstein condition guarantees the existence of the 2-form. Log-canonical singularities are precisely the singularities for which yP ? P s 0. In Corollary 4, we show that the previous theorem reduces to Hurwitz’s bound when X is a canonical cone over a nonhyperelliptic curve. We prove more generally: THEOREM ŽTheorem 3.. If Ž X, 0. is a cone o¨ er a smooth projecti¨ e cur¨ e C of genus g G 2, then GrCU ; AutŽ C .. Suppose further that Ž X, 0. is the cone o¨ er the canonical embedding C ; P gy 1. Then G is equal to the full group of graded automorphisms of the coordinate ring of Ž X, 0. and GrCU s AutŽ C .. Thus < GrCU < s 84Ž g y 1. whenever X is a canonical cone over a curve C such that
2. NOTATION Fix a normal surface singularity germ Ž X, 0.. Let p : Ž Y, E . ª Ž X, 0. be a good resolution, not necessarily minimal, and E s D Ei the exceptional set. Let K be a canonical divisor on Y. If Ž X, 0. is Gorenstein, then K is an integral divisor supported on E; otherwise K is Q-Cartier. Suppose
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P q N s K q E is the Zariski decomposition w11x of K q E; we briefly remind the reader of this decomposition. Both P and N are Q-divisors supported on E which satisfy the following properties: 1.
N is effective.
2.
P ? Ei s 0 if Ei is in the support of N.
3.
P ? Ei G 0 for all Ei .
The self-intersection number yP ? P, which is a nonnegative rational number, is a topological invariant of the link M s X l S 2 ny1. It can be computed directly from any good resolution dual graph G; the only ambiguities for this identification M ; G are in the cases of cusp and cyclic quotient singularities w9x. However, these are log-canonical ŽLC. singularities, for which our invariant vanishes anyway ŽyP ? P s 0 m K q E is effective.. The total list of log-canonical singularities is described in detail by Kawamata w6x. Briefly, they consist of simple elliptic singularities, cusp singularities, smooth points, and quotients of these by a finite group. Aside from topological invariant, yP ? P is a characteristic number ŽSection 3 or w12x. and is independent of the resolution of Ž X, 0.. For the remainder of the paper, Ž X, 0. ; ŽC n, 0. will denote the germ of a quasi-homogeneous Gorenstein surface singularity. We will assume throughout that Ž X, 0. is not log-canonical. Let R s Cw x 1 , . . . , x n xrJ be the graded coordinate ring of Ž X, 0.. The ideal J is generated by weighted homogeneous polynomials relative to weights, say, Ž q1 , . . . , qn .. Thus R s [R i has grading R i s f g R: f Ž t x. s t i f Žx.4 . If we assume that the weights are all positive, then the grading on the coordinate ring if equivalent to the existence of a good CU -action on X w10x. By the Gorenstein condition, the dualizing sheaf H 0 Ž X y 04 , V 2 . is a free graded R-module of rank one. Fix a graded generator v ; v is a nowhere zero holomorphic 2-form on Ž X y 04.. The degree d of v is the integer such that Ryd corresponds to C v in the dualizinig module R v . Equivalently, t v ª tydv under the CU-action. For non]log-canonical surfaces, d is strictly positive w14, Corollary 3.3x. Let G ; Aut Ž R . be the maximal reductive subgroup as in w5, 8, 13x, where every reductive subgroup of AutŽ R . is conjugate to a subgroup of G. Let G 0 ; G be the connected component of the identity. If, as in our case, Ž X, 0. is quasi-homogeneous and not a cyclic quotient singularity A n, q , then G 0 s CU .
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3. QUASI-HOMOGENEOUS SINGULARITIES As CU is in the center of G w13, Prop. 3.10x, G preserves the grading on R. To verify this, take f g R s and g g G. Under the CU -action, f Ž t¨ . s t s f Ž ¨ .; thus gf Ž t¨ . s gt s f Ž ¨ . s t s Ž gf .Ž ¨ .. G also preserves the grading on R v and, in particular, GŽC v . ; C v . Accordingly, there is a surjective morphism I: G ª CU which sends g to a g , where a g is defined by the property g v s a g v . Surjectivity follows from the fact that if g g CU then g v s gydv and d / 0. The group of symplectic automorphisms G s g g G: g v s v 4 is simply the kernel of I. Automorphisms preserving the grading on R are called equivariant automorphisms of Ž X, 0.. We will often use the notation G ; AutŽ R . and G ; AutŽ X, 0. interchangeably. In the following proposition consider G as a subgroup of the equivariant automorphisms acting on Ž X, 0.. LEMMA 1. If Ž X, 0. is a non]log-canonical, quasi-homogeneous Gorenstein surface singularity, then G has the following properties: Ža. G is finite. Žb. G acts freely on X y 04 . Žc. XrG is Gorenstein. Žd. < G < F 42ŽyP ? P .. Proof. Ža. The surjective mapping G ª GrCU has kernel Ž G l CU . , Z d Ž g g G l CU « g v s gydv s v .. Thus GrCU , GrZ d . The order of the group GrCU is simply the number of connected components of G. Thus G is finite with order < G < s d < GrCU < .
Ž 1.
One can see from here that Theorem 2 will be proven by Ž1. and part Žd. of the lemma. Žb. Suppose a nonidentity element g g G fixes a smooth point q on X. Let H be the finite cyclic subgroup generated by g. Take an H-invariant analytic neighborhood U of q and choose analytic coordinates Ž x, y . such that q s Ž0, 0.. The action of H can be linearized so that H ; GLŽ2. on U. Let a nonidentity element g g H be represented by the diagonal matrix Ž l01 l02 .. We claim that neither eigenvalue is equal to 1. If v s f Ž x, y . dx n dy, then g v s l1 l2 f Ž l1 x, l2 y . dx n dy s f Ž x, y . dx n dy since g fixes v . But f has a nonzero constant term, which forces l1 l2 s 1. Since g was chosen to be nontrivial, neither l i s 1, so q is an
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isolated fixed point for g. By restricting to a small enough neighborhood of the singular point 04 , G acts freely on X y 04 . Now consider the covering p : X ª XrG. Žc. It is well known that under our circumstances XrG is normal. It now remains to construct a nowhere zero holomorphic 2-form on XrG y 04 via p . For any q g XrG y 04 , let U be an evenly covered neighborn hood of q. Let Vi 4is1 be the slices of py1 ŽU .; for each i there exists an ; isomorphism h i : Vi ª U from X y 04 ª XrG y 04 . Define a local 2form, v XU , on U by letting v XU s ˙ h i Ž v < V i . for any i. This definition is independent of the choice of i for v is G-invariant. Cover XrG y 04 with such evenly covered neighborhoods Uq 4 and define a 2-form v X on XrG y 04 by the restrictions v X < Uq s ˙ v XUq. Clearly, the v XUq patch together since the v < V i patch on X y 04 . Žd. We quickly review two properties of the invariant yP ? P; proofs can be found respectively in w12x and w3x. The first was mentioned in Section 2: yP ? P is a characteristic number. In our context, this means, if Ž X, 0. ª Ž X X , 0. is an m-sheeted cover of normal surface singularities which is unramified off 04 , then yPX ? PX s mŽyPX X ? PX X .. The second property is that for all non]log-canonical Gorenstein singularities we have the inequality yP ? P G 1r42. Now by Ža., Žb., and Žc. we have just such a < G <-sheeted cover X ª XrG s X X of a Gorenstein singularity Ž X X , 0. Žand XrG is not log-canonical because X is not log-canonical.. Whence yPX ? PX s < G <ŽyPX X ? PX X . G < G <Ž1r42., from which it follows < G < F 42ŽyPX ? PX .. THEOREM 2. Let Ž X, 0. be a non]log-canonical, quasi-homogeneous Gorenstein surface singularity. If v is a fixed nowhere zero holomorphic 2-form on X y 04 of degree d, then < GrCU < F Ž42rd .ŽyP ? P .. Proof. Combine Ž1.: < G < s d < GrCU < with the inequality from Žd.: < G < F 42ŽyP ? P .. 4. HOMOGENEOUS SINGULARITIES Let Ž X, 0. be the cone over a smooth Žnondegenerate. projective curve C ; P ny 1 with homogeneous coordinate ring Cw x 1 , . . . , x n xrJ. If C is a nonhyperelliptic curve of genus g G 3, then it is well known that K C embeds C into P gy 1 and K C f OC Ž1.. In this case the image of C in P gy 1 is called the canonical cur¨ e. When Ž X, 0. is the cone over a canonical curve C, X s X Ž C, K C . is called the canonical cone. THEOREM 3. If Ž X, 0. is a cone o¨ er a smooth projecti¨ e cur¨ e C of genus g G 2, then GrCU ; AutŽ C .. When Ž X, 0. is the canonical cone, G is equal
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to the full group of graded automorphisms of the coordinate ring of Ž X, 0. and GrCU s AutŽ C .. Proof. Let R s Cw x 1 , . . . , x n xrJ be the coordinate ring of Ž X, 0. and H ; AutŽ R . the subgroup of all graded automorphisms. Viewing G ; H as equivariant automorphisms of Ž X, 0., the first statement of the theorem is verified by confirming the inclusion HrCU ; AutŽ C .. From the identification Ž X y 04.rCU s C, it follows that CU acts trivially on C and equivariant automorphisms of X leave C fixed, thereby inducing automorphisms on C. This defines a homomorphism
u : H ª Aut Ž C . , with CU ; kerŽ u .. To see that CU s kerŽ u ., let s g H. As the ideal J contains no linear polynomials, s is determined by its action on the C-module generated by Ž x 1 , . . . , x n .. Seeing that s is graded, it lifts to a linear automorphism sˆ of Cw x 1 , . . . , x n x Žan element of GLŽ n, C.. and hence to P ny 1. Now, if s g H is not in CU , then s cannot be in the kernel of u . For, sˆ g AutŽP ny 1 . m sˆ g PGLŽ n y 1, C., whence the fixed point locus of sˆ in P ny 1 consists of a union of points and linear subspaces. But C is nondegenerate in P ny 1 ; therefore we must have u Ž s . s sˆ < C acting nontrivially on C. Referring to the second assertion in the theorem, suppose X s X Ž C, K C . is a canonical cone and R s [G Ž C, K Cmn .. We establish that H s G and u is surjective. Since C is nonhyperelliptic, R is generated in degree one w2x. For surjectivity, any automorphism of C sends K C to itself and, consequently, lifts to graded automorphisms of R. By definition, G is the maximal reductive subgroup; hence the equality will follow from observing that H is also reductive. First note HrCU is reductive because it is finite. This makes H reductive: by general group theory H is the extension of the reductive group CU by the reductive group HrCU : 0 ª CU ª H ª HrCU ª 0. A corollary is Hurwitz’s bound in the nonhyperelliptic case. COROLLARY 4 ŽHurwitz.. Let C be a nonhyperelliptic cur¨ e of genus g G 3. Then
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can easily verify that K q E s yE, N s 0, and P s yE. Thus yP ? P s 2 g y 2. Plugging in the above,
5. EXAMPLES EXAMPLE 5. According to Theorem 3, if C is an curve which achieves Hurwitz’s bound Žand there are infinitely many of them w1, Theorem 5.7x and X Ž C, K C . is the canonical cone over C, then < GrCU < also takes on its bound. EXAMPLE 6. Conversely, Theorem 2 is not always ideal, especially in some simple situations. For example, suppose X is non]log-canonical and given by f Ž x, y, z . s x a q y b q z c, where Ž a, b, c . s 1 and 1ra q 1rb q 1rc - 1. The CU -action on X is t Ž x, y, z . ¬ Ž t b c x, t ac y, t ab z ., giving us weights Ž bc, ac, ab.. Then for non-RDP, quasi-homogeneous hypersurface singularities, 2
yP ? P s Ž abc y bc y ac y ab . rabc s Ž 1 y 1ra y 1rb y 1rc .
2
1 abc
by w12x Žor indirectly w15x.. Let v s Ž1rz cy 1 . dx n dy be a graded generator of H 0 Ž X y 04 , V 2 .; the degree of v is Ž abc y bc y ac y ab.. Thus, by Theorem 2, < GrCU < F Ž42.Ž1 y 1ra y 1rb y 1rc .. Though, in this case, we can do better, actually showing G s CU . To this end, any s g G preserves the weights, and so must be of the form su, ¨ , w : Ž x, y, z . ª Ž ux, ¨ y, wz ., where u, ¨ , w, m g CU and u a s ¨ b s w c s m. Taking t as below, s comes from the good CU -action t Ž x, y, z . ¬ Ž t b c x, t ac y, t ab z .. t s Ž u1r b c ¨ 1r ac w 1r ab . rm2r ab c . We remark here that the upper bound < GrCU < F Ž42.Ž1 y 1ra y 1rb y 1rc . found above assumes the actual order of one if Ž a, b, c . s Ž2, 3, 7. and Ž X, 0. is the D 2, 3, 7-triangle singularity Žthis is not surprising, as D 2, 3, 7 is the only singularity for which yP ? P attains its minimum nonzero lower bound of 1r42 w3x.. This leads us to Conjecture 7. If Ž X, 0. is not a cone, the D 2, 3, 7-triangle singularity is the unique singularity for which < GrCU < s Ž42rd .ŽyP ? P .. QUESTION. Suppose Ž X, 0. is Gorenstein of the form z n s f Ž x, y . and f is not a weighted polynomial Ž e¨ ery hypersurface is Gorenstein.. We ha¨ e no grading and thus no symplectic group < G < to di¨ ide out by. Howe¨ er, Z n acts as an automorphism group of order n; what is the best bound?
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REFERENCES 1. R. D. M. Accola, Strongly branched coverings of closed Riemann surfaces, Proc. Amer. Math. Soc. 26 Ž1970., 315]332. 2. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, ‘‘Geometry of Algebraic Curves,’’ Vol. 1, Springer-Verlag, New York, 1985. 3. F. Ganter, Properties of a topological invariant for Gorenstein singularities, Math. Z. 223 Ž1996., 411]419. 4. P. A. Griffiths, ‘‘Geometry of Algebraic Curves,’’ Vol. 1, Springer-Verlag, New York, 1985. 5. H. Hauser and G. Muller, Algebraic singularities have maximal reductive automorphism ¨ groups, Nagoya Math. J. 113 Ž1989., 181]186. 6. Y. Kawamata, Crepant blowing up of 3-dimensional canonical singularities and its applications to degenerations of surfaces, Ann. of Math. 127 Ž1988., 93]163. 7. H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 Ž1977.. 8. G. Muller, Symmetries of surface singularities, preprint. ¨ 9. W. Neumann, A calculus for applied plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 286 Ž1981., 299]344. 10. P. Orlik and P. Wagreich, Isolated singularities of algebraic surfaces with CU action, Ann. of Math. Ž2. 93 Ž1971., 205]228. 11. F. Sakai, Anticanonical models of rational surfaces, Math. Ann. 269 Ž1984., 389]410. 12. J. Wahl, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 Ž1990., 625]637. 13. J. Wahl, Derivations, automorphisms and deformations of quasi-homogeneous singularities, Proc. Sympos. Pure Math. 40 Ž1983.. 14. J. Wahl, The jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 Ž1987.. 15. K. Watanabe, On the plurigenera of normal isolated singularities I, Math. Ann. 250 Ž1980., 65]94. 16. G. Xiao, Bound on automorphisms of surfaces of general type I, Ann. of Math. 139 Ž1994., 51]57.
JOURNAL OF ALGEBRA ARTICLE NO.
Automorphism Groups for Quasi-homogeneous Gorenstein Surface Singularities Frieda M. Ganter U Department of Mathematics, CSV-Fresno, Fresno, California 93740 Communicated by D. A. Buchsbaum Received May 30, 1996
Let Ž X, 0. be a non]log-canonical, quasi-homogeneous surface singularity germ. And let G ; AutŽ X, 0. be the maximal reductive subgroup. In this paper we bound the order of GrCU by yP ? P, a purely topological invariant. Hurwitz’s theorem comes out as a corollary. We use the standard approach of taking the quotient space of Ž X, 0. by the action of a finite group of automorphisms acting freely on X y 04. Q 1998 Academic Press
Contents. 1. Introduction 2. Notation 3. Quasi-homogeneous singularities 4. Homogeneous singularities 5. Examples
1. INTRODUCTION Hurwitz’s theorem for a smooth projective curve of genus g G 2 bounds the order of the automorphism group,
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0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
GORENSTEIN SURFACE SINGULARITIES
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In the local situation, if Ž X, 0. is a germ of a quasi-homogeneous Gorenstein surface singularity, with graded coordinate ring R, then we bound the order of GrCU , where G ; AutŽ R . is the maximal reductive subgroup ŽSection 3.. The bound we find, which is essentially yP ? P, is a topological invariant of Ž X, 0.. This is the content of the following theorem: THEOREM ŽTheorem 2.. Let Ž X, 0. be a non]log-canonical, quasihomogeneous Gorenstein surface singularity. Suppose there is a fixed nowhere zero holomorphic 2-form on X y 04 of degree d. Then < GrCU < F Ž42rd .ŽyP ? P .. Note that the Gorenstein condition guarantees the existence of the 2-form. Log-canonical singularities are precisely the singularities for which yP ? P s 0. In Corollary 4, we show that the previous theorem reduces to Hurwitz’s bound when X is a canonical cone over a nonhyperelliptic curve. We prove more generally: THEOREM ŽTheorem 3.. If Ž X, 0. is a cone o¨ er a smooth projecti¨ e cur¨ e C of genus g G 2, then GrCU ; AutŽ C .. Suppose further that Ž X, 0. is the cone o¨ er the canonical embedding C ; P gy 1. Then G is equal to the full group of graded automorphisms of the coordinate ring of Ž X, 0. and GrCU s AutŽ C .. Thus < GrCU < s 84Ž g y 1. whenever X is a canonical cone over a curve C such that
2. NOTATION Fix a normal surface singularity germ Ž X, 0.. Let p : Ž Y, E . ª Ž X, 0. be a good resolution, not necessarily minimal, and E s D Ei the exceptional set. Let K be a canonical divisor on Y. If Ž X, 0. is Gorenstein, then K is an integral divisor supported on E; otherwise K is Q-Cartier. Suppose
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P q N s K q E is the Zariski decomposition w11x of K q E; we briefly remind the reader of this decomposition. Both P and N are Q-divisors supported on E which satisfy the following properties: 1.
N is effective.
2.
P ? Ei s 0 if Ei is in the support of N.
3.
P ? Ei G 0 for all Ei .
The self-intersection number yP ? P, which is a nonnegative rational number, is a topological invariant of the link M s X l S 2 ny1. It can be computed directly from any good resolution dual graph G; the only ambiguities for this identification M ; G are in the cases of cusp and cyclic quotient singularities w9x. However, these are log-canonical ŽLC. singularities, for which our invariant vanishes anyway ŽyP ? P s 0 m K q E is effective.. The total list of log-canonical singularities is described in detail by Kawamata w6x. Briefly, they consist of simple elliptic singularities, cusp singularities, smooth points, and quotients of these by a finite group. Aside from topological invariant, yP ? P is a characteristic number ŽSection 3 or w12x. and is independent of the resolution of Ž X, 0.. For the remainder of the paper, Ž X, 0. ; ŽC n, 0. will denote the germ of a quasi-homogeneous Gorenstein surface singularity. We will assume throughout that Ž X, 0. is not log-canonical. Let R s Cw x 1 , . . . , x n xrJ be the graded coordinate ring of Ž X, 0.. The ideal J is generated by weighted homogeneous polynomials relative to weights, say, Ž q1 , . . . , qn .. Thus R s [R i has grading R i s f g R: f Ž t x. s t i f Žx.4 . If we assume that the weights are all positive, then the grading on the coordinate ring if equivalent to the existence of a good CU -action on X w10x. By the Gorenstein condition, the dualizing sheaf H 0 Ž X y 04 , V 2 . is a free graded R-module of rank one. Fix a graded generator v ; v is a nowhere zero holomorphic 2-form on Ž X y 04.. The degree d of v is the integer such that Ryd corresponds to C v in the dualizinig module R v . Equivalently, t v ª tydv under the CU-action. For non]log-canonical surfaces, d is strictly positive w14, Corollary 3.3x. Let G ; Aut Ž R . be the maximal reductive subgroup as in w5, 8, 13x, where every reductive subgroup of AutŽ R . is conjugate to a subgroup of G. Let G 0 ; G be the connected component of the identity. If, as in our case, Ž X, 0. is quasi-homogeneous and not a cyclic quotient singularity A n, q , then G 0 s CU .
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3. QUASI-HOMOGENEOUS SINGULARITIES As CU is in the center of G w13, Prop. 3.10x, G preserves the grading on R. To verify this, take f g R s and g g G. Under the CU -action, f Ž t¨ . s t s f Ž ¨ .; thus gf Ž t¨ . s gt s f Ž ¨ . s t s Ž gf .Ž ¨ .. G also preserves the grading on R v and, in particular, GŽC v . ; C v . Accordingly, there is a surjective morphism I: G ª CU which sends g to a g , where a g is defined by the property g v s a g v . Surjectivity follows from the fact that if g g CU then g v s gydv and d / 0. The group of symplectic automorphisms G s g g G: g v s v 4 is simply the kernel of I. Automorphisms preserving the grading on R are called equivariant automorphisms of Ž X, 0.. We will often use the notation G ; AutŽ R . and G ; AutŽ X, 0. interchangeably. In the following proposition consider G as a subgroup of the equivariant automorphisms acting on Ž X, 0.. LEMMA 1. If Ž X, 0. is a non]log-canonical, quasi-homogeneous Gorenstein surface singularity, then G has the following properties: Ža. G is finite. Žb. G acts freely on X y 04 . Žc. XrG is Gorenstein. Žd. < G < F 42ŽyP ? P .. Proof. Ža. The surjective mapping G ª GrCU has kernel Ž G l CU . , Z d Ž g g G l CU « g v s gydv s v .. Thus GrCU , GrZ d . The order of the group GrCU is simply the number of connected components of G. Thus G is finite with order < G < s d < GrCU < .
Ž 1.
One can see from here that Theorem 2 will be proven by Ž1. and part Žd. of the lemma. Žb. Suppose a nonidentity element g g G fixes a smooth point q on X. Let H be the finite cyclic subgroup generated by g. Take an H-invariant analytic neighborhood U of q and choose analytic coordinates Ž x, y . such that q s Ž0, 0.. The action of H can be linearized so that H ; GLŽ2. on U. Let a nonidentity element g g H be represented by the diagonal matrix Ž l01 l02 .. We claim that neither eigenvalue is equal to 1. If v s f Ž x, y . dx n dy, then g v s l1 l2 f Ž l1 x, l2 y . dx n dy s f Ž x, y . dx n dy since g fixes v . But f has a nonzero constant term, which forces l1 l2 s 1. Since g was chosen to be nontrivial, neither l i s 1, so q is an
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isolated fixed point for g. By restricting to a small enough neighborhood of the singular point 04 , G acts freely on X y 04 . Now consider the covering p : X ª XrG. Žc. It is well known that under our circumstances XrG is normal. It now remains to construct a nowhere zero holomorphic 2-form on XrG y 04 via p . For any q g XrG y 04 , let U be an evenly covered neighborn hood of q. Let Vi 4is1 be the slices of py1 ŽU .; for each i there exists an ; isomorphism h i : Vi ª U from X y 04 ª XrG y 04 . Define a local 2form, v XU , on U by letting v XU s ˙ h i Ž v < V i . for any i. This definition is independent of the choice of i for v is G-invariant. Cover XrG y 04 with such evenly covered neighborhoods Uq 4 and define a 2-form v X on XrG y 04 by the restrictions v X < Uq s ˙ v XUq. Clearly, the v XUq patch together since the v < V i patch on X y 04 . Žd. We quickly review two properties of the invariant yP ? P; proofs can be found respectively in w12x and w3x. The first was mentioned in Section 2: yP ? P is a characteristic number. In our context, this means, if Ž X, 0. ª Ž X X , 0. is an m-sheeted cover of normal surface singularities which is unramified off 04 , then yPX ? PX s mŽyPX X ? PX X .. The second property is that for all non]log-canonical Gorenstein singularities we have the inequality yP ? P G 1r42. Now by Ža., Žb., and Žc. we have just such a < G <-sheeted cover X ª XrG s X X of a Gorenstein singularity Ž X X , 0. Žand XrG is not log-canonical because X is not log-canonical.. Whence yPX ? PX s < G <ŽyPX X ? PX X . G < G <Ž1r42., from which it follows < G < F 42ŽyPX ? PX .. THEOREM 2. Let Ž X, 0. be a non]log-canonical, quasi-homogeneous Gorenstein surface singularity. If v is a fixed nowhere zero holomorphic 2-form on X y 04 of degree d, then < GrCU < F Ž42rd .ŽyP ? P .. Proof. Combine Ž1.: < G < s d < GrCU < with the inequality from Žd.: < G < F 42ŽyP ? P .. 4. HOMOGENEOUS SINGULARITIES Let Ž X, 0. be the cone over a smooth Žnondegenerate. projective curve C ; P ny 1 with homogeneous coordinate ring Cw x 1 , . . . , x n xrJ. If C is a nonhyperelliptic curve of genus g G 3, then it is well known that K C embeds C into P gy 1 and K C f OC Ž1.. In this case the image of C in P gy 1 is called the canonical cur¨ e. When Ž X, 0. is the cone over a canonical curve C, X s X Ž C, K C . is called the canonical cone. THEOREM 3. If Ž X, 0. is a cone o¨ er a smooth projecti¨ e cur¨ e C of genus g G 2, then GrCU ; AutŽ C .. When Ž X, 0. is the canonical cone, G is equal
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to the full group of graded automorphisms of the coordinate ring of Ž X, 0. and GrCU s AutŽ C .. Proof. Let R s Cw x 1 , . . . , x n xrJ be the coordinate ring of Ž X, 0. and H ; AutŽ R . the subgroup of all graded automorphisms. Viewing G ; H as equivariant automorphisms of Ž X, 0., the first statement of the theorem is verified by confirming the inclusion HrCU ; AutŽ C .. From the identification Ž X y 04.rCU s C, it follows that CU acts trivially on C and equivariant automorphisms of X leave C fixed, thereby inducing automorphisms on C. This defines a homomorphism
u : H ª Aut Ž C . , with CU ; kerŽ u .. To see that CU s kerŽ u ., let s g H. As the ideal J contains no linear polynomials, s is determined by its action on the C-module generated by Ž x 1 , . . . , x n .. Seeing that s is graded, it lifts to a linear automorphism sˆ of Cw x 1 , . . . , x n x Žan element of GLŽ n, C.. and hence to P ny 1. Now, if s g H is not in CU , then s cannot be in the kernel of u . For, sˆ g AutŽP ny 1 . m sˆ g PGLŽ n y 1, C., whence the fixed point locus of sˆ in P ny 1 consists of a union of points and linear subspaces. But C is nondegenerate in P ny 1 ; therefore we must have u Ž s . s sˆ < C acting nontrivially on C. Referring to the second assertion in the theorem, suppose X s X Ž C, K C . is a canonical cone and R s [G Ž C, K Cmn .. We establish that H s G and u is surjective. Since C is nonhyperelliptic, R is generated in degree one w2x. For surjectivity, any automorphism of C sends K C to itself and, consequently, lifts to graded automorphisms of R. By definition, G is the maximal reductive subgroup; hence the equality will follow from observing that H is also reductive. First note HrCU is reductive because it is finite. This makes H reductive: by general group theory H is the extension of the reductive group CU by the reductive group HrCU : 0 ª CU ª H ª HrCU ª 0. A corollary is Hurwitz’s bound in the nonhyperelliptic case. COROLLARY 4 ŽHurwitz.. Let C be a nonhyperelliptic cur¨ e of genus g G 3. Then
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FRIEDA M. GANTER
can easily verify that K q E s yE, N s 0, and P s yE. Thus yP ? P s 2 g y 2. Plugging in the above,
5. EXAMPLES EXAMPLE 5. According to Theorem 3, if C is an curve which achieves Hurwitz’s bound Žand there are infinitely many of them w1, Theorem 5.7x and X Ž C, K C . is the canonical cone over C, then < GrCU < also takes on its bound. EXAMPLE 6. Conversely, Theorem 2 is not always ideal, especially in some simple situations. For example, suppose X is non]log-canonical and given by f Ž x, y, z . s x a q y b q z c, where Ž a, b, c . s 1 and 1ra q 1rb q 1rc - 1. The CU -action on X is t Ž x, y, z . ¬ Ž t b c x, t ac y, t ab z ., giving us weights Ž bc, ac, ab.. Then for non-RDP, quasi-homogeneous hypersurface singularities, 2
yP ? P s Ž abc y bc y ac y ab . rabc s Ž 1 y 1ra y 1rb y 1rc .
2
1 abc
by w12x Žor indirectly w15x.. Let v s Ž1rz cy 1 . dx n dy be a graded generator of H 0 Ž X y 04 , V 2 .; the degree of v is Ž abc y bc y ac y ab.. Thus, by Theorem 2, < GrCU < F Ž42.Ž1 y 1ra y 1rb y 1rc .. Though, in this case, we can do better, actually showing G s CU . To this end, any s g G preserves the weights, and so must be of the form su, ¨ , w : Ž x, y, z . ª Ž ux, ¨ y, wz ., where u, ¨ , w, m g CU and u a s ¨ b s w c s m. Taking t as below, s comes from the good CU -action t Ž x, y, z . ¬ Ž t b c x, t ac y, t ab z .. t s Ž u1r b c ¨ 1r ac w 1r ab . rm2r ab c . We remark here that the upper bound < GrCU < F Ž42.Ž1 y 1ra y 1rb y 1rc . found above assumes the actual order of one if Ž a, b, c . s Ž2, 3, 7. and Ž X, 0. is the D 2, 3, 7-triangle singularity Žthis is not surprising, as D 2, 3, 7 is the only singularity for which yP ? P attains its minimum nonzero lower bound of 1r42 w3x.. This leads us to Conjecture 7. If Ž X, 0. is not a cone, the D 2, 3, 7-triangle singularity is the unique singularity for which < GrCU < s Ž42rd .ŽyP ? P .. QUESTION. Suppose Ž X, 0. is Gorenstein of the form z n s f Ž x, y . and f is not a weighted polynomial Ž e¨ ery hypersurface is Gorenstein.. We ha¨ e no grading and thus no symplectic group < G < to di¨ ide out by. Howe¨ er, Z n acts as an automorphism group of order n; what is the best bound?
GORENSTEIN SURFACE SINGULARITIES
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