EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS

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1 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS DAVID SWINARSKI Abstract. We say a Riemann surface has a large automorphism group if # Aut(C) > 4(g 1). We find equations of all genus 4 and 5 Riemann surfaces C with large automorphism groups, and all but two of the genus 6 Riemann surfaces with large automorphisms groups. The strategy used is based on the Eichler trace formula and the projection formula from the character theory of finite groups. It is not completely algorithmic, but surprisingly yields enough information that we can finish each example by hand. In the last section we discuss how an algorithm for computing flattening stratifications could make the last step algorithmic as well. Every finite group G arises as a group of automorphisms of some smooth compact Riemann surface S (or complex algebraic curve C) or as the full automorphism group of a punctured Riemann surface [3, 7]. However, the property of #G being large compared to the genus g of S is special. A general curve has no nontrivial automorphisms at all. There is the well-known bound due to Hurwitz: #G 84(g 1), but most curves with automorphisms do not even come close to this. For instance, in genus 3, the Hurwitz bound is 168, but only five curves have automorphism groups satisfying #G > 4. We will call an automorphism group large if #G > 4(g 1). By Riemann-Hurwitz, this implies that the quotient curve is P 1, and the quotient morphism branches over 3 or 4 points. Using Fuchsian group theory, it is possible to set up computer searches for large automorphism groups. Algebraists have carried this out [,1] and published lists of pairs (G, g) such that there is a curve of genus g with large automorphism group G, but we don t know equations of all these curves. In other words, there is no published algorithm yet which given the group produces equations of the curve. In this note I will outline a non-algorithmic strategy for producing equations of Riemann surfaces with automorphisms under their canonical embeddings (or for curves, an equation of the form y = f(x) instead). Surprisingly, although my strategy is non-algorithmic, the algorithmic steps generally get us close enough to the answer that we can finish them by hand. Section presents an example which is very typical in this sense. In particular, I am able to find all of the genus 4 and 5 Riemann surfaces with large automorphism groups, and all but two of the genus 6 Riemann surfaces with large automorphism groups. Many of these example are classically known, but others may be new, and to my knowledge, this is the first time they are all collected in one place. In Section 1 I outline the strategy in more detail. In Section I apply the strategy to one example; this section serves both as an illustration of the strategy and a mini-tutorial on how to use my Magma code. Section 3 is intended to be a reference work. In the final section, Section 4, I will discuss an idea for how the final step could be made algorithmic, too; in theory, this could lead to an algorithm for finding equations of any Riemann surface with automorphisms. (The missing ingredient is knowing how to compute a so-called flattening stratification.) 0.1. Acknowledgements. I would like to thank the undergraduates from my Introductory VIGRE Research Group held in Fall 010 at the University of Georgia: Eddie Beck, Zachary Freeland, Tyler Johnson, Malik Obeidin, Jacob Rooney, and Lev Tovstopyat-Nelip, where I began the main programming of this work was begun. I am also very grateful to the computational algebra group at Date: August 8,

2 DAVID SWINARSKI the University of Sydney for hosting me for a visit, where a great deal of this work was completed. A large number of people have provided supplied tips, conversations, and encouragement over a number of years, especially Marston Conder, Johan de Jong, Amy Ksir, Ian Morrison, Jen Paulhus, Roy Smith, Robert Varley, and John Voight. 0.. Online. My webpage for this project is davids/ivrg/results.html This page contains links to the latest version of my Magma code, the main tables below, and several files of notes on specific examples. 1. The strategy Input: a finite group G, the genus g, and the ramification data (e 1,..., e r ) of the quotient morphism C P 1. For 4 g 10 this data is available in [1]. Marston Conder has also published a list on his website going to g = 101; however, this includes many groups which are not the full automorphism group of a Riemann surface. Step 1. Compute the classes and character table of G, and find a set of surface kernel generators. The notion of surface kernel generators has turned out to be very useful in the study of curves with automorphisms. Definition 1.1 A set of surface kernel generators for the group G and ramification data (e 1,..., e r ) is a sequence of elements M i G such that (1) M i : i = 1,..., r = G; () Order(M i ) = e i ; and (3) M 1 M M r = 1. We will use the surface kernel generators for Steps, 3 and 4 below. Step. Test for ity. There are two reasons why it is useful to know early on whether the curve is. First, if C is, then the canonical divisor does not define an embedding. Second, curves can be specified by an equation of the form y = f(x), and their equations are well understood [13, 14]. So it is useful to know this gonality in advance, so that we don t try to find canonical equations when we shouldn t, and so that we can take advantage of what is known about curves. The curve C is if and only if G contains a central involution σ with g + fixed points. The number of fixed points can be computed using Lemma 11.5 of [] (or Theorem 7 of [8]): Theorem 1. Let σ be an automorphism of order h > 1 of a Riemann surface X of genus g. Let (M 1,..., M r ) be a set of surface kernel generators for X. Let Fix X,u (σ) be the set of fixed points of X where σ acts on a neighborhood of the fixed point by z exp(πiu/h)z. Then Fix X,u (σ) = C G (σ) M i s.t. h m i σ M m i u/h i Thus, we can find all the central involutions in G, and compute the number of fixed points for each one (if there are any). If the curve is, we turn to [13] to get the equation of the form y = f(x), and then we can turn to [14] if we want to get the equations of C undering a linear series. If the curve is not, we proceed. 1 m i

3 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 3 Step 3: Test for ity. If C is non-, then its canonical divisor defines an embedding, and the canonical ideal is generated at worst by quadrics and cubics. More specifically, by Petri s Theorem, the canonical ideal of a general curve is generated by quadrics, and it is only for curves and plane quintics that additional cubic generators are needed. So it is helpful to know in advance whether the curve is. Costa and Izquierdo give criteria for testing for ity [4, Prop. 1,]. Step 4: Use the Eichler Trace Formula to compute the character of the action of G on S 1 as well as on I and I 3. Proposition 1.3 (Eichler Trace Formula [5]) Suppose g C, and let σ be a nontrivial automorphism of C of order h. Write χ m for the character of the representation of Aut(C) on H 0 (C, mk). Then 1 + ζh u Fix C,u (σ) 1 ζ u if m = 1 1 u<h h (u,h)=1 χ m (σ) = Fix C,u (σ) ζu(m%h) h 1 ζh u if m 1 u<h (u,h)=1 Let S = Sym H 0 (C, K). By Noether s Theorem, the sequence 0 I m S m H 0 (C, mk) 0 is exact for all m 1, and so we can compute the character of the action of G on I and I 3 from knowing χ 1, χ, χ 3, Sym χ 1 and Sym 3 χ 1. Step 5: Test whether C is a plane quintic. I don t know a necessary and sufficient condition for testing for plane quintics. However, these only occur in genus 6, and then the canonical model lies on the Veronese surface in P 5. One way to detect plane quintics, then, is to see whether H 0 (C, K) = Sym V for some three-dimensional (not necessarily irreducible) represention V of G. In practice, this sufficed when studying all the genus 6 curves with large automorphism groups. Step 6: Obtain matrix generators for the action of G on H 0 (C, K). In a few cases, Magma was unable to find matrix generators for the required representation. I believe this is an implementation error, not an algorithmic one. In these cases, I was able to find the needed representation in [10], [11], or by using GAP. Step 7: Use the projection formula to obtain candidate quadrics (and cubics, if applicable). We use the projection formula (cf. e.g. [6, Formula (.3)]): the projection of V onto its isotypical component V m i i is given by π i = dim V i 1 G χ Vi (g)g. At this stage, the process becomes non-algorithmic. In general, an irreducible representation V i will show up with multiplicity in S or S 3. Frequently, some of the copies of V i will belong to I or I 3 while others will belong to H 0 (C, K) or H 0 (C, 3K), and I don t know an algorithmic way to decide how to split V m i i between them. However, at this stage, we may proceed following our intuition from algebraic geometry and frequently still obtain the equations we seek. In the next section we work through an example. g G Here is an example which is very typical.. An example: genus 5, G = (3, 8)

4 4 DAVID SWINARSKI There is a one-parameter family of Riemann surfaces of genus 5 whose full automorphism group is the group G = (3, 8) in the notation of the GAP library of small finite groups. The quotient morphism branches over 4 points of P 1 with ramification (,,, 4). Magma V.17-6 Mon Jul :08:35 on dopey [Seed = ] Type? for help. Type <Ctrl>-D to quit. > G:=SmallGroup(3,8); > G; GrpPC : G of order 3 = ^5 PC-Relations: G.^ = G.4, G.^G.1 = G. * G.4, G.3^G.1 = G.3 * G.5 The algorithmic steps described in the section above have been implemented in a function RunExample() in Magma. This function returns the images of the generators G.1 G.5 under the representation G H 0 (C, K) as well as lists of candidate quadrics and cubics. We can run the example at hand via > MatGens38,Q,C:=RunExample(G,5,[,,,4]); First, Magma computes the conjugacy classes and character table of G. (Since there is no canonical way to order the classes or the characters, we do this at the beginning to fix an ordering once and for all.) Conjugacy Classes of group G [1] Order 1 Length 1 Rep Id(G) [] Order Length 1 Rep G.5 [3] Order Length 1 Rep G.4 [4] Order Length 1 Rep G.4 * G.5 [5] Order Length Rep G.3 [6] Order Length Rep G.3 * G.4 [7] Order Length 4 Rep G.1 * G. [8] Order Length 4 Rep G.1 [9] Order 4 Length Rep G. * G.3 * G.5 [10] Order 4 Length Rep G. * G.3

5 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 5 [11] Order 4 Length Rep G. * G.5 [1] Order 4 Length Rep G. [13] Order 4 Length 4 Rep G.1 * G.3 [14] Order 4 Length 4 Rep G.1 * G. * G.3 Character Table of Group G Class Size Order p = X X X X X X X X X X X X X *I *I X *I -*I Explanation of Character Value Symbols I = RootOfUnity(4) Next, Magma computes a set of surface kernel generators. Since by [1] we know that this curve is unique, it is only necessary to study one set of surface kernel generators. Magma then tests whether C is. SKGs: [ G.3 * G.4, G.1 * G.4, G.1 * G. * G.4, G. * G.3 ] Is? false Next, Magma computes the G-module structure of several relevant G-modules. We list this information by the multiplicities of the irreducible representation (as ordered in the character table), i.e. if V = V m Vr mr, then the vector of multiplicities (m 1,..., m r ) is displayed below: Multiplicities of irreducibles in relevant G-modules:

6 6 DAVID SWINARSKI I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] S_1 =[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ] H^0(C,1K)=[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 ] I_ =[ 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] S_ =[,, 0, 0, 0,, 1, 0, 0, 1, 1, 1, 0, 1 ] H^0(C,K)=[ 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1 ] I_3 =[ 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, ] S_3 =[ 0, 0,,, 1, 0, 0,, 5, 1, 1, 1,, 4 ] H^0(C,3K)=[ 0, 0, 1, 1, 1, 0, 0, 1,, 1, 1, 1, 1, ] ItimesS1=[ 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, ] Is clearly not generated by quadrics? false In the last line, Magma has compared I 3 and I S 1. If the multiplicity of any irreducible in I 3 exceeds its multiplicity in I S 1, then we cannot have I 3 I S 1, and so the curve must either be or a plane quintic. Remark: It would be interesting to know if the converse is true i.e. if C is or a plane quintic and #G > 4(g 1), then is there always an irreducible representation V i whose multiplicity in I 3 exceeds its multiplicity in I S 1? This would give a test for gonality that, to my knowledge, is new. Next Magma finds matrix generators for the representation ρ : G H 0 (C, K). First, it returns the list ρ(g.i), where G.i is a generator of the group G in whatever format Magma has it stored. Next, it returns ρ(m i ) for the surface kernel generators M i : Matrix generators for action on H^0(C,K): Field K Cyclotomic Field of order 3 and degree 16 [ [ ] [ ] [ ] [ ] [ ], [ ] [ ] [ ] [ ] [ ], [ ] [ ] [ ] [ z^8] [ z^8 0] ] Surface Kernel Generators: [ [ ] [ ] [ ] [ z^8] [ z^8 0], [ ] [ ] [ ]

7 ] EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 7 [ ] [ ], [ ] [ ] [ ] [ ] [ ], [ ] [ ] [ ] [ z^8 0] [ z^8] Next, Magma uses the projection formula from representation theory to find the quadrics (and cubics, if relevant) which may be in I and I 3 : Finding quadrics: I contains a 1-dimensional subspace of CharacterRow 1 Dimension Multiplicity [ a^, b^ + c^ ] I contains a 1-dimensional subspace of CharacterRow Dimension Multiplicity [ b*c, d^ - e^ ] I contains a 1-dimensional subspace of CharacterRow 6 Dimension Multiplicity [ b^ - c^, d*e ] Thus, we seek coefficients µ 1,..., µ 6 such that I C = µ 1 a + µ (b + c ), µ 3 bc + µ 4 (d e ), µ 5 (b c ) + µ 6 (de). At this point, the algorithmic steps are done. I don t know an algorithmic way to choose the necessary one-dimensional subspaces of these three two-dimensional subspaces. However, we can proceed intuitively. The canonical ideal of a curve should contain no monomials. In fact, let us suppose that none of the coefficients µ i is zero. We can divide each equation by the coefficient of the first term to set µ 1 = µ 3 = µ 5 = 1: I C = a + µ (b + c ), bc + µ 4 (d e ), (b c ) + µ 6 (de)

8 8 DAVID SWINARSKI Next, note that by scaling a, we may assume that µ = 1, and by scaling d and e, we may assume that µ 4 = 1. Thus we obtain I C = a + b + c, bc + d e, b c + µ 6 (de). Now µ 6 is the only remaining unknown coefficient. However, we are expecting a 1-dimensional family of curves with this automorphism group, and we have obtained a candidate pencil. The last step is to check that some member of this pencil is a smooth curve. If there is, then by Zariski-openness of smoothness, a general member of this pencil is smooth. > K<z>:=CyclotomicField(3); > P4<a,b,c,d,e>:=ProjectiveSpace(K,4); > mu1:=1; > C:=Scheme(P4,[a^+b^+c^,b*c+d^-e^,b^-c^+mu1*d*e]); > Dimension(X); 1 > IsSingular(X); false As a final check, we have Magma test that the matrix generators of G H 0 (C, K) are automorphisms of C by running commands of the following type: > Automorphism(C,MatGens38[1]); Mapping from: Sch: C to Sch: C with equations : -a b -c e d and inverse -a b -c e d (Magma would return an error if this matrix did not define an automorphism of C.) 3. Results In the following tables, we list the equations of genus 4 and 5 curves with large automorphism groups, as well as the genus 6 curves with large automorphism groups whose quotient morphisms branch over 3 points of P 1. The following notation is used for the tables. In [1] the letter δ denotes the dimension of the family of curves; if the quotient morphism branches over 3 points of P 1, then δ = 0, and if it branches over 4 points of P 1, then δ = 1 (we can vary the cross-ratio of the 4 points). In the first column of each of the following tables, we list: the number assigned to this curve in [1], the GAP identifier of the finite group (the first number in this pair is the order of the group), the ramification data of the quotient morphism, and whether the curve is,, or a plane quintic. Surface kernel generators are given in many cases as a g-tuple denoting the images of the variables under the automorphism. For example, the 4-tuple ( c, b, a, d) associated to the genus 4 curve with automorphism group (7, 4) encodes the automorphism a c, b b, c a, d d.

9 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 9 Figure 1. Genus 4, δ = 0 Data Equations Surface kernel generators 1 (10,34) = S 5,4,5 (7,4),3,1 3 (7,40),4,6 4 (40,8),4,10 5 (36,1),6,6 6 (3,19),4,16 7 (4,3) 3,4,6 8 (18,) (,9,18) 9 (15,1) = Z 15 3,5,15 ad + bc, a c ab + bd c d a + b + c abc d 3 ab + cd, a 3 b 3 + c 3 d 3 y = x 10 1 ad + c, a 3 d 3 + b 3 y = x 9 x y = x(x 4 1)(x 4 + tx + 1), t = 3 y = x 9 1 b + cd, a 3 + bc + d 3 first generator to be added (z 4 b, z 3 d, z a, zc); 5 ) ( c, b, a, d), ( w c, w a, w b, wd); w = exp( πi 3 ) ( a, b, d, c), ( c, d, w b, wa); w = exp( πi 3 ) ( zd, b, c, (z 1)a), ( d, (z 1)b, zc, ( z + 1)a); 6 ) (z10 a, z5 b, z5 c, z5 d), (z 3 a, z 6 b, z 9 c, z 3 d); 15 )

10 10 DAVID SWINARSKI Figure. Genus 4, δ = 1 Data Equations Surface kernel generators 10 (36,10),,,3 ab + cd, a 3 b 3 + t(c 3 d 3 ) (wb, w a, c, d), ( a, b, d, c), ( b, a, wd, w c); w = exp( πi 3 ) 11 (4,1)=S 4,,,4 1 (0,4),,,5 13 (18,3),,3,3 14 (16,7),,,8 a + b + c + td, abc d 3 y = (x 5 t 5 )(x 5 t 5 ) ac b, a 3 b 3 + d 3 + tabc y = x(x 4 t 4 )(x 4 t 4 ) ( a, c, b, d), (c, b, a, d), ( a, c, b, d) (w b, wa, c, d), (b, a, c, d), (wa, wb, wc, w d); w = exp( πi 3 )

11 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 11 Figure 3. Genus 5, δ = 0 Data Equations Surface kernel generators 1 a ibd, ( c, 1 b + 1 d, a, i )b + 1 d, e), (19,181) c + b + d, (ic, z3,3,8 e + ib id b + z5 d, e, z7 )b + z5 d, ia); 8 ) (160,34),4,5 3 (10,35),3,10 4 (96,195),4,6 5 (64,3),4,8 6 (48,14),4,1 7 (48,30) 3,4,4 8 (40,5),4,0 9 (30,),6,15 10 (,),11, a + b + c + d + e, a + (z 3 + z )c + ( z 3 z )d e, b c + ( z 3 z )d + (z 3 + z )e ; 5 ) y = x 11 11x 6 x c + ce + d de + e, a + c + ( 4z + )cd + zce + d zde +(z 1)e, b zc + ( z + 4)cd ce zd +de + (z 1)e ; 6 ) b + c + d + e, a + b + c d e, b c id + ie y = x 1 1 y = x 1 33x 8 33x y = x 11 x a + be, ac + de, ad bc, ab + c d + e 3, ae + cd + b 3 y = x 11 1 ( c, d, a, b, e), ( a, c, b, e, d) ( b, a, d e, c + e, e) ((z 1)b, za, d, c, c e); 6 ) ( a, b, c, d, e), (ia, e, d, b, c) ( a, z 9 e, z 3 d, z 1 c, z 6 b), ( z 10 a, z 7 e, z 14 d, z 11 c, z 13 b); 15 )

12 1 DAVID SWINARSKI Figure 4. Genus 5, δ = 1 Data Equations Surface kernel generators a + ab + b + c + cd +ce + d 11 + de + e, a (48,48) b + t(c ( a, b, d, c, e), +cd de e,,,3 ( a + b, b, c, d, e), ), ab + b + t( c (b, a, c + e, d + e, e) ce + d + de) 1 (3,43),,,4 13 (3,8),,,4 14 (3,7),,,4 15 (4,14),,,6 16 (4,8),,,6 17 (4,13),,3,3 18 (0,4),,,10 a + be + cd, b + c + tde, d + e + tbc a + b + c, bc + d e, b c + tde a bc, a + d + e, b c + tde y = (x 6 t 6 )(x 6 t 6 ) a + b b c + c, ab + t(b + bc c ) + d + e, ac + t(b bc c ) +zd + ( z + 1)e ; 6 ) tab + c + d + e, a + c + (z 1)d ze, b + c zd + (z 1)e ; 6 ) y = x(x 5 t 5 )(x 5 t 5 ) ( a, d, e, b, c), ( a, ze, zd, z 3 c, z 3 b), ( a, z e, z d, z c, z b); 8 ) ( a, b, c, ie, id), ( a, b, c, e, d), ( a, c, b, d, e) ( a, b, c, e, d), ( a, ic, ib, d, e), ( a, c, b, d, e) ( a, b, c, d, e), ( a, b c, c, e, d), ( a, b, b c, ze, (z 1)d); 6 ) ( a, b, c, d, e), ( a, b, c, d, e), ((z 1)a, zb, c, d, e); 6 )

13 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 13 Figure 5. Genus 6, δ = 0 Data Equations Surface kernel generators 1 (150,5),3,10 g 6 (10,34),4,6 3 (7,15),4,9 4 (56,7),4,14 5 (48,6),4,4 6 (48,9) = GL (3),6,8 7 (48,15),6,8 8 (39,1),3,13 g 6 9 (30,1),10,15 g 6 10 (6,) = Z,13,6 11 (1,) = Z 3,7,1 x 5 + y 5 + z 5 Normalization of the Wiman sextic; equations of this curve are known (see e.g. [9]), but I have not successfully run my strategy on this curve yet y 3 = x x y = x 14 1 y = x 13 x y = x(x 4 1)(x x 4 + 1)) y 3 = x 8 1 x 4 y + y 4 z + z 4 x a 5 + b 4 c + zb 3 c + z b c 3 + z 3 bc 4 ; 5 ) y = x 13 1 y 3 = x 7 1 (a, z 4 c, zb), (z 3 a, zb, z b + zc); 5 )

14 14 DAVID SWINARSKI 4. Can we use flattening stratifications to make the last step algorithmic? 4.1. A one-dimensional family of genus 6 curves. Consider the following example. It is known that the group S 4 is the automorphism group of a 1-dimensional family of genus 6 curves. The quotient curve is P 1, and the quotient morphism branches at 4 points with ramifications indices,, 3, 4. Moreover, these curves are neither nor. Let C be a member of this family. We can use the Eichler trace formula to find the action of S 4 on H 0 (C, mk) for m = 1,. Moreover, we can find matrix generators for the action on H 0 (C, K): ,, , and we can decompose the action on quadrics in these variables to get F 1 := a + ab ac + b bc + c + d + df + e + ef + f ; F := µ 1 (a ac b + bc) + µ (ad ae af bd be 3bf + ce + cf) + µ 3 (8de + 4df + 4ef + f ) F 3 := µ 1 (ab + ac + 1 b 1 c ) + µ ( 1 ad 1 ae + 1 af + bd + be + 3/bf 3/cd 1 ce cf) + µ 3 (d de + e f ) F 4 := µ 4 (a c ) + µ 5 (ad + bd be cd) + µ 6 (d + df e ef) F 5 := µ 4 (ab ac bc + c ) + µ 5 ( 1 ad 1 ae 1 af 1 bd + 1 be + cd + 1 cf) + µ 6 ( d df + e + ef) F 6 := µ 4 (b c ) + µ 5 (ad + ae + af + bd + bf cd cf) + µ 6 (d e ef f ) Let S = P P, where the first copy of P has coordinates [µ 1 : µ : µ 3 ], and the second copy has coordinates [µ 4 : µ 5 : µ 6 ]. For each s S, let I s be the ideal F 1,... F 6. We are guaranteed that there exists at least one s S such that I s defines a smooth curve in P 5. How can we find such a point s? Note that most values of s define empty subschemes of P 5. It is a theorem (due originally to either Mumford or Grothendieck, I believe) that there exists a flattening stratification of S. The following statements are taken from the secondary source [15]: Definition 4.1 Let F be a coherent sheaf on P n S. A flattening stratification for F over S is a finite disjoint collection {S i } of locally closed subschemes of S such that (1) set-theoretically, S S i ; () for any morphism g : T S, the pullback (1 g) F is flat over T if and only if each g i : S i T is open and closed in T.,

15 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 15 Proposition 4. ([15, Prop. 7.]) Let F be a coherent sheaf on P n S. There exists a flattening stratification {S P (z) } for F, indexed by numerical polynomials P (z), such that for all g : T S, we have: F T is T -flat with Hilbert polynomial P (z) if and only if g factors through S P (z). Returning to the example above: suppose we knew how to compute a flattening stratification of S. Then we could find the stratum S P (z) where P (z) = (g )z g+1 = 10z 5. This stratum may be irreducible and disconnected, but it should contain at least one irreducible component where the generic fiber is a smooth curve in P 5. In fact, we could use our ability to compute flattening stratifications a second time: let J be the ideal sheaf defined by J = I + Jac(I); for each s S, J s computes the singular locus of I s. Then we can compute a flattening stratification {T j } for J, and find the stratum R 0 corresponding to the Hilbert polynomial P (z) = 0. Then any closed point of R 0 S 10z 5 should give us the equations we seek. So, how can we compute flattening stratifications? I don t know the answer, but I have had some suggestions: (1) Fitting ideals are often useful to compute flattening stratifications, but this requires finitely presentation. () Dave Bayer pointed me to [1], which studies the behavior of Gröbner bases with respect to extension of scalars. (3) Mark Watkins suggested a different approach: search over finite fields for points s S where the fiber dimension is correct, then try to lift. I close with an example that suggests a method for computing flattening stratifications. 4.. Example: a flattening stratification for the twisted cubic. Let S = P 1 with coordinates s, t. Let P 3 S have coordinates a,b,c,d. Consider the ideal I = s a c t b, s a d t b c, s b d t c in P 3 S. Let X = V (I). I want to think of X as the total space of a one-dimensional family of closed subschemes of P 3. For s = t, we get the twisted cubic. Experimentally, for generic [s : t], it seems we get a subscheme with Hilbert polynomial P (t) = 8, given by two fat points, each of multiplicity 4, in P 3. How can we see this algorithmically? Can we stratify the base S into locally closed subsets S i such that over S i, the map f : X S, with fibers having Hilbert polynomial P i (t)? Which Hilbert polynomials can arise? Here s the calculation that finally got the wheels turning in my brain. Compute a Groebner basis for I over Q[s, t]. In Macaulay, I did i1 : A=QQ[s,t]; i : B=A[a..d]; i3 : I=ideal(s*a*c-t*b^,s*a*d-t*b*c,s*b*d-t*c^); i4 : flatten entries gens gb I o4 : {t*c^-s*b*d, t*b*c-s*a*d, t*b^-s*a*c, s*(s-t)*a*c*d, s*d*(b^-a*c), s*(s-t)*a*b*d, s*a*(c^-b*d), s^*(s-t)*a^*d^} (Confession: that s not literally the Macaulay output, I factored some of the Groebner basis elements a little.) Let s call these polynomials g 1,..., g 8. Now, whenever the coefficients appearing in g 1,..., g 8 are all units (that is, if s, t, and s t are units), then we will get the initial ideal c, b c, b, a c d, b d, a b d, a c, a d, and hence we should expect to get the same Hilbert function over the open set A st(s t). On the other hand, if s = 0 or if s = t, then we will lose some of the generators g 4,..., g 8 entirely, and that could potentially lead to a different Hilbert function and polynomial. Also, if t = 0, we won t lose any generators, but we will get different leading terms in g 1, g, g 3, so maybe we could get a different Hilbert function in this case, too (and indeed we do). A direct calculation shows that for (s, t) = (0, 1), (1, 1), or (1, 0), the Hilbert polynomial of the ideal

16 16 DAVID SWINARSKI t c s b d, t b c s a d, t b s a c, s (s t) a c d, s d (b a c), s (s t) a b d, s a (c b d), s (s t) a d is P (t) = 3 t + 1. So I think the flattening stratification for this example should be: S 1 = P 1 {0, 1, }, labeled with Hilbert polynomial P 1 (t) = 8; S = {[0 : 1], [1 : 1], [1 : 0]}, labeled with Hilbert polynomial P (t) = 3 t Proposed generalization. The example above suggests the following generalization. Let f : X S be a morphism. For simplicity, assume S = Spec(A) is reduced. Suppose X P n S is defined by an ideal I that is homogeneous in the variables x 0,..., x n with coefficients in A. Compute a Groebner basis G for I. (Question: does the term order matter? Can we ignore this if we take a universal Groebner basis instead?) Write each generator g i as g i = n i j=1 c i,jx e j, where c i,j is in A, and x e j is a monomial. Let T = {(i, j)} be the set of all index pairs. Collect all the coefficients c i,j of all the terms appearing in G. Let F = T c i,j. Then A F should be an open set where f is flat. Now, start knocking out combinations of c i,j s. That is, for each subset U of T, consider the scheme S U defined by S U = {c i,j = 0 : (i, j) U}. Two subsets U, U could potentially lead to the same Hilbert polynomial, we can collect all the S U s that carry fibers with the same Hilbert polynomial into one stratum labeled by that polynomial, to get our stratification. Remarks. (1) Here I ve only considered a special case of flattening stratifications; I think you can start with any coherent sheaf, not necessarily an ideal sheaf. This special case is what I need for my application. Can it be generalized? () This also gives (I think) a very hands-on proof of semicontinuity of fiber dimension. You can see in the construction above that there s a big open set where the fibers all have dimension d. Then, as you start knocking out generators, the dimensions of the schemes defined by the monomial ideals could grow (since you knocked out generators), but this only happens over progressively smaller locally closed sets. References [1] Dave Bayer, André Galligo, and Mike Stillman, Gröbner bases and extension of scalars, Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp MR15399 (94j:1301) 15 [] Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 80, Cambridge University Press, Cambridge, 000. MR (00i:14034) 1, [3] W. Burnside, Theory of groups of finite order, Dover Publications Inc., New York, d ed. MR (16,1086c) 1 [4] Antonio F. Costa and Milagros Izquierdo, Maximal order of automorphisms of Riemann surfaces, J. Algebra 33 (010), no. 1, 7 31, DOI /j.jalgebra MR56486 (011a:14063) 3 [5] H. M. Farkas and I. Kra, Riemann surfaces, nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 199. MR (93a:30047) 3 [6] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 19, Springer- Verlag, New York, A first course; Readings in Mathematics. MR (93a:0069) 3 [7] Leon Greenberg, Conformal transformations of Riemann surfaces, Amer. J. Math. 8 (1960), MR01988 (3 #A319) 1 [8] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. () 17 (1966), MR00169 (34 #1511) [9] Naoki Inoue and Fumiharu Kato, On the geometry of Wiman s sextic, J. Math. Kyoto Univ. 45 (005), no. 4, MR668 (007m:14078) 13 [10] Izumi Kuribayashi and Akikazu Kuribayashi, Automorphism groups of compact Riemann surfaces of genera three and four, J. Pure Appl. Algebra 65 (1990), no. 3, 77 9, DOI / (90)90107-S. MR10785 (9a:30041) 3

17 EQUATIONS OF RIEMANN SURFACES OF GENUS 4, 5, AND 6 WITH LARGE AUTOMORPHISM GROUPS 17 [11] Akikazu Kuribayashi and Hideyuki Kimura, Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, , DOI / (90) MR (91j:30033) 3 [1] K. Magaard, T. Shaska, S. Shpectorov, and H. Völklein, The locus of curves with prescribed automorphism group, Sūrikaisekikenkyūsho Kōkyūroku 167 (00), , available at arxiv:math.ag/ Communications in arithmetic fundamental groups (Kyoto, 1999/001). MR ,, 5, 8 [13] Tanush Shaska, Determining the automorphism group of a curve, Proceedings of the 003 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 003, pp (electronic), DOI / , (to appear in print). MR03519 (005c:14037) [14] Jan Stevens, Deformations of singularities, Lecture Notes in Mathematics, vol. 1811, Springer-Verlag, Berlin, 003. MR (004b:3049) [15] Stein Arild Strømme, Elementary introduction to representable functors and Hilbert schemes, Parameter spaces (Warsaw, 1994), Banach Center Publ., vol. 36, Polish Acad. Sci., Warsaw, 1996, pp MR (98m:14004) 14, 15 Software Packages Referenced [16] The GAP Group, GAP: Groups, Algorithms, and Programming, a system for computational discrete algebra (008), available at Version [17] Dan Grayson and Mike Stillman, Macaulay : a software system for research in algebraic geometry (010), available at Version 1.4. [18] School of Mathematics and Statistics Computational Algebra Research Group University of Sydney, MAGMA computational algebra system (008), available at Version address: davids@math.uga.edu