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Progress in MathematicsVolume 282

Series EditorsHyman BassJoseph OesterleAlan Weinstein

Cohomologicaland Geometric Approachesto Rationality Problems

New Perspectives

Fedor BogomolovYuri TschinkelEditors

BirkhauserBoston • Basel • Berlin

Fedor Bogomolov Yuri TschinkelDepartment of Mathematics Department of MathematicsNew York University New York UniversityCourant Institute Courant Institute

of Mathematical sciences of Mathematical sciencesNew York, NY 10012 New York, NY 10012U.S.A [email protected] [email protected]

Library of Congress Control Number:

Mathematics Subject Classification (2000): 11R32, 12F12, 13A50, 14D20,14E05, 14E08, 14F20, 14G05,14G15, 14H10, 14H45, 14H60, 14J32, 14J35, 14L30

c© Birkhauser Boston, a part of Springer Science+Business Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media LLC, 233 SpringStreet, New York, NY 10013, USA), except for brief excerpts in connection with reviews or schol-arly analysis. Use in connection with any form of information storage and retrieval, electronic adapta-tion, computer software, or by similar or dissimilar methodology now known or hereafter developed isforbidden.The use in this publication of trade names, trademarks, service marks and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

Printed on acid-free paper.

Birkhauser Boston is part of Springer Science+Business Media (www.birkhauser.com)

2009939069

ISBN 978-0-8176-4933-3 e-ISBN 978-0-8176-4934-0DOI 10.1007/978-0-8176-4934-0

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

The Rationality of Certain Moduli Spaces of Curves ofGenus 333Ingrid Bauer and Fabrizio Catanese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Rationality of the Moduli Space of Curves of Genus 333 afterP. KatsyloChristian Bohning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Unramified Cohomology of Finite Groups of Lie TypeFedor Bogomolov, Tihomir Petrov and Yuri Tschinkel . . . . . . . . . . . . . . . . . 55

Sextic Double SolidsIvan Cheltsov and Jihun Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Moduli Stacks of Vector Bundles on Curves and the King–Schofield Rationality ProofNorbert Hoffmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Noether’s Problem for Some ppp-GroupsShou-Jen Hu and Ming-chang Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Generalized Homological Mirror Symmetry andRationality QuestionsLudmil Katzarkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

The Bogomolov Multiplier of Finite Simple GroupsBoris Kunyavskiı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

vi Contents

Derived Categories of Cubic FourfoldsAlexander Kuznetsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Fields of Invariants of Finite Linear GroupsYuri G. Prokhorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

The Rationality Problem and Birational RigidityAleksandr V. Pukhlikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Preface

The rationality problem links algebra and geometry. On the level of fields itcan be formulated as follows: Let K be a field of finite transcendence degreeover a ground field k, i.e., K is the field of rational functions on an algebraicvariety defined over k. Decide whether or not K is a purely transcendentalextension of k. It became apparent that, for k = C, the geometry of thecorresponding variety is tightly linked with rationality of the field K.

The difficulty of the rationality problem depends on the transcendencedegree of K over k or, geometrically, on the dimension of the variety. A majorsuccess of the 19th and the first half of the 20th century algebraic geometrywas a complete solution of the rationality problem in dimensions one and twoover algebraically closed ground fields of characteristic zero. In these cases itsuffices to consider finite, easily computable geometric invariants. For suchfields in dimension two, the presence of sufficiently flexible rational curvessuffices for the rationality of the field. This geometric property (rational con-nectedness) defines a class of algebraic varieties (and fields) which in higherdimensions is substantially larger than the class of rational varieties: it isknown that most of rationally connected varieties are not rational already indimension three. It is still an open question whether or not they are unira-tional, i.e., the corresponding function fields are proper subfields of rationalfields.

The papers in this collection are devoted to various aspects of rationality.Four articles address the rationality of quotient spaces. The classical pre-Hilbert 19th century invariant theory dealt mostly with rings of invariants oflinear actions of groups on complex linear spaces—many beautiful explicit con-structions can be traced back to the works of Sylvester, Cayley, and Gordan.Emmy Noether, who was a student of Paul Gordan, one of the experts in thetheory of explicit invariants, formulated a general question: are all such fieldsof invariants rational? The first counterexamples, over nonclosed ground fields,were constructed by Swan, and over the complex numbers, by Saltman. Thismade it even more interesting to determine all rational fields of invariants.

viii Preface

The article by Prokhorov reviews known results in case of four-dimensionalcomplex representations of finite groups. It is shown that almost all of thearising quotients are rational. The paper by Hu and Kang contains a proof ofrationality for quotients by small p-groups. Kunyavskiı proves triviality of thefirst cohomological obstruction to rationality, for all finite simple groups andquasi-simple groups of Lie type. He also shows the existence of exceptions—there are quasi-simple groups which are central extensions of one particularsimple group of Lie type with a nonvanishing obstruction. Remarkably, theseare exactly the extensions which are not of Lie type themselves. The paper byBogomolov, Petrov, and Tschinkel is devoted to more general cohomologicalinvariants of quasi-simple groups of Lie type. It is shown that all obstructionsto rationality coming from nonramified cohomology vanish for a large classof quasi-simple groups of Lie type, at least when the group of coefficients hasorder coprime to the characteristic of the field of definition of the group. Twopapers treat rationality of the moduli space of curves of genus three. This wasestablished 15 years ago by Katsylo but the proof was computationally veryinvolved and difficult to follow. C. Bohning wrote a detailed and transparentversion of this proof. Bauer and Catanese prove rationality of the relatedmoduli space of curves of genus three with a fixed nontrivial 3-torsion point inthe Jacobian. They identify this moduli space with the moduli space of curvesof bidegree (4, 4) ⊂ P

1 × P1, with simple singularities at the presecribed set

of six points. Hoffmann’s paper is devoted to rationality of moduli spaces ofvector bundles on curves. It contains a stack-theoretic version of the King–Schofield proof of the rationality of the moduli space of vector bundles on aprojective curve having a fixed determinant line bundle, with coprime rankand degree of the determinant.

Rational fields have many automorphisms and the absence of these pointsto nonrationality. Pukhlikov surveys nonrationality results obtained via thestudy of rigidity properties of the projective models of the field. First examplesof rigid rationally connected varieties were discovered by Manin and Iskovskikhin the seventies. Cheltsov and Park consider an arithmetic aspect of ratio-nality: a classical diophantine problem concerns the description of rationalsolutions of a system of polynomial equations, i.e., rational points on alge-braic varieties. If the variety is rational or unirational, then there are manysuch points, at least over a finite extension of the ground field. This potentialdensity of rational points holds for all Fano varieties of dimension ≤ 3 withan exception of one class: double covers of the three-dimensional projectivespace ramified over a surface of degree 6. Cheltsov–Park show that potentialdensity of rational points holds if the ramification surface has at least onesingular point. They also show that these varieties are superrigid.

Katzarkov introduces a completely novel approach to the rationality prob-lem, based on ideas of Mirror Symmetry. The Mirror Symmetry conjectureis still open but the ideas suggested by physics already highlight new inter-esting phenomena related to rationality. Applications of derived categories torationality questions are discussed in the paper by Kuznetsov. He studies the

Preface ix

derived category of coherent sheaves of four-dimensional cubic hypersurfaces.It is expected that a generic cubic fourfold is nonrational; this remains one ofthe challenging problems in the field.

Needless to say, the book does not cover the whole spectrum of modernapproaches to rationality. Nevertheless, we hope that it provides a glimpse ofthe variety of related concrete problems, new methods and results.

Fedor BogomolovYuri Tschinkel

April 2009

The Rationality of Certain Moduli Spacesof Curves of Genus 3

Ingrid Bauer and Fabrizio Catanese

Mathematisches InstitutUniversität Bayreuth, NW IID-95440 Bayreuth, [email protected], [email protected]

Summary. We prove rationality of the moduli space of pairs of curves of genusthree together with a point of order three in their Jacobian.

Key words: Rationality, moduli spaces of curves

2000 Mathematics Subject Classification codes: 14E08, 14H10, 14H45

1 Introduction

The aim of this paper is to give an explicit geometric description of the bira-tional structure of the moduli space of pairs (C, η), where C is a general curveof genus 3 over an algebraically closed field k of arbitrary characteristic andη ∈ Pic0(C)3 is a nontrivial divisor class of 3-torsion on C.

As was observed in [B-C04, Lemma (2.18)], if C is a general curve ofgenus 3 and η ∈ Pic0(C)3 is a nontrivial 3-torsion divisor class, then wehave a morphism ϕη := ϕ|KC+η| × ϕ|KC−η| : C → P

1 × P1, corresponding

to the sum of the linear systems |KC + η| and |KC − η|, which is birationalonto a curve Γ ⊂ P

1 × P1 of bidegree (4, 4). Moreover, Γ has exactly six

ordinary double points as singularities, located in the six points of the setS := {(x, y)|x �= y, x, y ∈ {0, 1,∞}}.

In [B-C04] we only gave an outline of the proof (and there is also a minorinaccuracy). Therefore we dedicate the first section of this article to a detailedgeometrical description of such pairs (C, η), where C is a general curve of genus3 and η ∈ Pic0(C)3 \ {0}.

The main result of the first section is the following:

Theorem 1.1. Let C be a general (in particular, nonhyperelliptic) curve ofgenus 3 over an algebraically closed field k (of arbitrary characteristic) andη ∈ Pic0(C)3 \ {0}.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_1, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

2 I. Bauer and F. Catanese

Then the rational map ϕη : C → P1 × P

1 defined by

ϕη := ϕ|KC+η| × ϕ|KC−η| : C → P1 × P

1

is a morphism, birational onto its image Γ , which is a curve of bidegree (4, 4)having exactly six ordinary double points as singularities. We can assume, upto composing ϕη with a transformation of P

1 × P1 in PGL(2, k)2, that the

singular set of Γ is the set

S := {(x, y) ∈ P1 × P

1|x �= y ;x, y ∈ {0, 1,∞}}.

Conversely, if Γ is a curve of bidegree (4, 4) in P1×P

1, whose singularitiesconsist of exactly six ordinary double points at the points of S, its normaliza-tion C is a curve of genus 3, such that OC(H2 −H1) =: OC(η) (where H1,H2 are the respective pullbacks of the rulings of P

1 × P1) yields a nontrivial

3-torsion divisor class, and OC(H1) ∼= OC(KC + η), OC(H2) ∼= OC(KC − η).

From Theorem 1.1 it follows that

M3,η := {(C, η) : C is a general curve of genus 3, η ∈ Pic0(C)3 \ {0}}

is birational to P(V (4, 4,−S))/S3, where

V (4, 4,−S) := H0(OP1×P1(4, 4)(−2∑

a�=b,a,b∈{∞,0,1}(a, b))).

In fact, the permutation action of the symmetric group S3 := S({∞, 0, 1})extends to an action on P

1, so S3 is naturally a subgroup of PGL(2, k). Weconsider then the diagonal action of S3 on P

1 × P1, and observe that S3 is

exactly the subgroup of PGL(2, k)2 leaving the set S invariant. The actionof S3 on V (4, 4,−S) is naturally induced by the diagonal inclusion S3 ⊂PGL(2, k)2 .

On the other hand, if we consider only the subgroup of order three ofPic0(C) generated by a nontrivial 3-torsion element η, we see from Theo-rem 1.1 that we have to allow the exchange of η with −η, which correspondsto exchanging the two factors of P

1 × P1. Therefore

M3,〈η〉 := {(C, 〈η〉) : C general curve of genus 3, 〈η〉 ∼= Z/3Z ⊂ Pic0(C)}

is birational to P(V (4, 4,−S))/(S3 ×Z/2), where the action of the generatorσ (of Z/2Z) on V (4, 4,−S) is induced by the action on P

1 × P1 obtained by

exchanging the two coordinates.Our main result is the following:

Theorem 1.2. Let k be an algebraically closed field of arbitrary characteristic.We have:

1) the moduli space M3,η is rational;2) the moduli space M3,〈η〉 is rational.

Rationality of Moduli 3

One could obtain the above result abstractly from the method of Bogo-molov and Katsylo (cf. [B-K85]), but we prefer to prove the theorem whileexplicitly calculating the field of invariant functions. It mainly suffices to de-compose the vector representation of S3 on V (4, 4,−S) into irreducible fac-tors. Of course, if the characteristic of k equals two or three, it is no longerpossible to decompose the S3-module V (4, 4,−S) as a direct sum of irre-ducible submodules. Nevertheless, we can write down the field of invariantsand see that it is rational.

Acknowledgment. The research of the authors was performed in the realm ofthe DFG Forschergruppe 790 “Classification of algebraic surfaces and compactcomplex manifolds.”

2 The geometric description of pairs (C, η)

In this section we give a geometric description of pairs (C, η), where C is ageneral curve of genus 3 and η is a nontrivial element of Pic0(C)3, and weprove Theorem 1.1.

Let k be an algebraically closed field of arbitrary characteristic. We recallthe following observation from [B-C04, p. 374].

Lemma 2.1. Let C be a general curve of genus 3 and η ∈ Pic0(C)3 a non-trivial divisor class (i.e., η is not linearly equivalent to 0). Then the linearsystem |KC + η| is base point free. This holds more precisely under the as-sumption that the canonical system |KC | does not contain two divisors of theform Q + 3P , Q + 3P ′, and where the 3-torsion divisor class P − P ′ is theclass of η. This condition for all such η is in turn equivalent to the fact thatC is either hyperelliptic or it is nonhyperelliptic but the canonical image Σ ofC does not admit two inflexional tangents meeting in a point Q of Σ.

Proof. Note that P is a base point of the linear system |KC + η| if and only if

H0(C,OC(KC + η)) = H0(C,OC(KC + η − P )).

Since dimH0(C,OC(KC + η)) = 2 this is equivalent to

dimH1(C,OC(KC + η − P )) = 1.

Since H1(C,OC(KC + η−P )) ∼= H0(C,OC(P − η))∗, this is equivalent tothe existence of a point P ′ such that P − η ≡ P ′ (note that we denote linearequivalence by the classical notation “≡”). Therefore 3P ≡ 3P ′ and P �= P ′,whence in particular H0(C,OC(3P )) ≥ 2. By Riemann–Roch we have

dimH0(C,OC(KC − 3P )) =

deg(KC − 3P ) + 1− g(C) + dimH0(C,OC(3P )) ≥ 1.


In particular, there is a point Q such that Q ≡ KC − 3P ≡ KC − 3P ′.

Going backwards, we see that this condition is not only necessary, butsufficient. If C is hyperelliptic, then Q+ 3P,Q+ 3P ′ ∈ |KC |, hence P, P ′ areWeierstrass points, whence 2P ≡ 2P ′, hence P −P ′ yields a divisor class η of2-torsion, contradicting the nontriviality of η.

Consider now the canonical embedding of C as a plane quartic Σ. Ourcondition means, geometrically, that C has two inflection points P , P ′, suchthat the tangent lines to these points intersect in Q ∈ C.

We shall show now that the (nonhyperelliptic) curves of genus 3 whosecanonical image is a quartic Σ with the above properties are contained in afive-dimensional family, whence are special in the moduli spaceM3 of curvesof genus 3.

Let now p, q, p′ be three noncollinear points in P2. The quartics in P

2

form a linear system of dimension 14. Imposing that a plane quartic containsthe point q is one linear condition. Moreover, the condition that the linecontaining p and q has intersection multiplicity equal to 3 with the quarticin the point p gives three further linear conditions. Similarly for the point p′,and it is easy to see that the above seven linear conditions are independent.Therefore the linear subsystem of quartics Σ having two inflection points p, p′,such that the tangent lines to these points intersect in q ∈ Σ, has dimension14−3−3−1 = 7. The group of automorphisms of P

2 leaving the three pointsp, q, p′ fixed has dimension 2 and therefore the above quartics give rise to afive-dimensional algebraic subset of M3.

Finally, if the points P, P ′, Q are not distinct, we have (w.l.o.g.) P = Qand a similar calculation shows that we have a family of dimension 7− 3 = 4.

�

Consider now the morphism

ϕη(:= ϕ|KC+η| × ϕ|KC−η|) : C → P1 × P

1,

and denote by Γ ⊂ P1 × P

1 the image of C under ϕη.

Remark 2.2.1) Since η is nontrivial, either Γ is of bidegree (4, 4), or degϕη = 2 and Γ

is of bidegree (2, 2). In fact, deg ϕη = 4 implies η ≡ −η.2) We shall assume in the following that ϕη is birational, since otherwise

C is either hyperelliptic (if Γ is singular) or C is a double cover of an ellipticcurve Γ (branched in 4 points).

In both cases C lies in a five-dimensional subfamily of the moduli spaceM3 of curves of genus 3.

Let P1, . . . , Pm be the (possibly infinitely near) singular points of Γ , andlet ri be the multiplicity in Pi of the proper transform of Γ . Then, denotingby H1, respectively H2, the divisors of a vertical, respectively of a horizontal


line in P1 × P

1, we have that Γ ∈ |4H1 + 4H2 −∑mi=1 riPi|. By adjunction,

the canonical system of Γ is cut out by |2H1 + 2H2 −∑mi=1(ri − 1)Pi|, and

therefore

4 = degKC = Γ · (2H1 + 2H2 −m∑

i=1

(ri − 1)Pi) = 16−m∑

i=1

ri(ri − 1).

Hence∑mi=1 ri(ri − 1) = 12, and we have the following possibilities:

m (r1, . . . , rm)i) 1 (4)ii) 2 (3,3)iii) 4 (3,2,2,2)iv) 6 (2,2,2,2,2,2)

We will show now that for a general curve only the last case occurs, i.e.,Γ has exactly 6 singular points of multiplicity 2.

We denote by S the blowup of P1 × P

1 in P1, . . . , Pm, and let Ei be theexceptional divisor of the first kind, total transform of the point Pi.

We shall first show that the first case (i.e., m = 1) corresponds to the caseη ≡ 0.

Proposition 2.3. Let Γ ⊂ P1×P

1 be a curve of bidegree (4, 4) having a pointP of multiplicity 4, such that its normalization C ∈ |4H1 + 4H2 − 4E| hasgenus 3 (here, E is the exceptional divisor of the blowup of P

1 × P1 in P ).

ThenOC(H1) ∼= OC(H2) ∼= OC(KC).

In particular, if Γ = ϕη(C) (i.e., we are in the case m = 1), then η ≡ 0.

Remark 2.4. Let Γ be as in the proposition. Then the rational map P1 ×

P1 �� P

2 given by |H1 +H2−E| maps Γ to a plane quartic. Vice versa, givena plane quartic C′, blowing up two points p1, p2 ∈ (P1 × P

1) \ C′, and thencontracting the strict transform of the line through p1, p2, yields a curve Γ ofbidegree (4, 4) having a singular point of multiplicity 4.

Proof (of the proposition). Let H1 be the full transform of a vertical linethrough P . Then there is an effective divisor H ′

1 on the blowup S of P1 × P

1

in P such that H1 ≡ H ′1 +E. Since H1 ·C = E ·C = 4, H ′

1 is disjoint from C,whence OC(H1) ∼= OC(E). The same argument for a horizontal line throughP obviously shows that OC(H2) ∼= OC(E). If h0(C,OC(H1)) = 2, then thetwo projections p1, p2 : Γ → P

1 induce the same linear series on C, thusϕ|H1| and ϕ|H2| are related by a projectivity of P

1, hence Γ is the graph of aprojectivity of P

1, contradicting the fact that the bidegree of Γ is (4, 4).Therefore we have a smooth curve of genus 3 and a divisor of degree 4

such that h0(C,OC(H1)) ≥ 3. Hence h0(C,OC(KC −H1)) ≥ 1, which impliesthat KC ≡ H1. Analogously, KC ≡ H2. �


The next step is to show that for a general curve C of genus 3, cases ii)and iii) do not occur. In fact, we show:

Lemma 2.5. Let C be a curve of genus 3 and η ∈ Pic0(C)3\{0} such that ϕηis birational and the image ϕη(C) = Γ has a singular point P of multiplicity3. Then C belongs to an algebraic subset of M3 of dimension ≤ 5.

Proof. Let S again be the blow up of P1×P

1 in P , and denote by E the excep-tional divisor. Then OC(E) has degree 3 and arguing as in Proposition 2.3, wesee that there are points Q1, Q2 on C such that OC(Hi) ∼= OC(Qi+E). There-fore OC(Q2−Q1) ∼= OC(H2−H1) ∼= OC(KC−η−(KC+η)) ∼= OC(η), whence3Q1 ≡ 3Q2, Q1 �= Q2. This implies that there is a morphism f : C → P

1 ofdegree 3, having double ramification in Q1 and Q2. By Hurwitz’ formula thedegree of the ramification divisor R is 10 and since R ≥ Q1+Q2 f has at mosteight branch points in P

1. Fixing three of these points to be∞, 0, 1, we obtain(by Riemann’s existence theorem) a finite number of families of dimension atmost 5. �

From now on, we shall make the following

Assumptions.C is a curve of genus 3, η ∈ Pic0(C)3 \ {0}, and

1) |KC + η| and |KC − η| are base point free;2) ϕη : C → Γ ⊂ P

1 × P1 is birational;

3) Γ ∈ |4H1 +4H2| has only double points as singularities (possibly infinitelynear).

Remark 2.6. By the considerations so far, we know that a general curve ofgenus 3 fulfills the assumptions for any η ∈ Pic0(C)3 \ {0}.

We use the notation introduced above: we have π : S → P1 × P

1 andC ⊂ S, C ∈ |4H1 + 4H2 − 2

∑6i=1 Ei|.

Remark 2.7. Since S is a regular surface, we have an easy case of Ramanu-jam’s vanishing theorem: if D is an effective divisor which is 1-connected (i.e.,for every decomposition D = A+B with A,B > 0, we have A ·B ≥ 1), thenH1(S,OS(−D)) = 0.

This follows immediately from Ramanujam’s lemma ensuringH0(D,OD) =k, and from the long exact cohomology sequence associated to

0→ OS(−D)→ OS → OD → 0.

In most of our applications we shall show that D is linearly equivalentto a reduced and connected divisor (this is a stronger property than 1-connectedness).


We know now that OC(H1 +H2) ∼= OC(2KC), i.e.,

OC ∼= OC(3H1 + 3H2 −6∑

i=1

2Ei).

Since h1(S,OS(−H1 −H2)) = 0, the exact sequence

0→ OS(−H1 −H2)→ OS(3H1 + 3H2 −6∑

i=1

2Ei)

→ OC(3H1 + 3H2 −6∑

i=1

2Ei) ∼= OC → 0, (1)

is exact on global sections.In particular, h0(S,OS(3H1 + 3H2 −

∑6i=1 2Ei)) = 1. We denote by G

the unique divisor in the linear system |3H1 + 3H2 −∑6i=1 2Ei|. Note that

C ∩G = ∅ (since OC ∼= OC(G)).

Remark 2.8. There is no effective divisor G on S such that G = G + Ei,since otherwise G · C = −2, contradicting that G and C have no commoncomponent.

This means that G + 2∑6i=1 Ei is the total transform of a curve G′ ⊂

P1 × P

1 of bidegree (3,3).

Lemma 2.9. h0(G,OG) = 3, h1(G,OG) = 0.

Proof. Consider the exact sequence

0→ OS(KS)→ OS(KS +G)→ OG(KG)→ 0.

Since h0(S,OS(KS)) = h1(S,OS(KS)) = 0, we get

h0(S,OS(KS +G)) = h0(G,OG(KG)).

Now, KS + G ≡ H1 + H2 −∑6

i=1 Ei, therefore (KS + G) · C = −4, whenceh0(G,OG(KG)) ∼= h0(S,OS(KS +G)) = 0.

Moreover, h1(G,OG(KG)) = h1(S,OS(KS + G)) + 1, and by Riemann–Roch we infer that, since h1(S,OS(KS + G)) = h0(S,OS(−G)) = 0, thath1(S,OS(KS +G)) = 2. �

We will show now that G is reduced, hence, by the above lemma, we shallobtain that G has exactly three connected components.

Proposition 2.10. G is reduced.


Proof. By Remark 2.8 it is sufficient to show that the image of G in P1 × P

1,which we denoted by G′, is reduced.

Assume that there is an effective divisor A′ on P1×P

1 such that 3A′ ≤ G′.We clearly have A′∩Γ �= ∅ but, after blowing up the six points P1, . . . , P6, thestrict transforms of A′ and of Γ are disjoint, whence A′ and G′ must intersectin one of the Pi’s, contradicting Remark 2.8.

If G′ is not reduced, we may uniquely write G′ = 2D1 + D2 with D1, D2

reduced and having no common component. Up to exchanging the factors ofP

1 × P1, we have the following two possibilities:

i) D1 ∈ |H1 +H2|;ii) D1 ∈ |H1|.In the first case also D2 ∈ |H1 + H2| and its strict transform is disjoint fromC. Remark 2.8 implies that D2 meets Γ in points which do not belong to D1,whence D2 has double points where it intersects Γ . Since D2 · Γ = 8 we seethat D2 has two points of multiplicity 2, a contradiction (D2 has bidegree(1, 1)).

Assume now that D1 ∈ |H1|. Then, since 2D1 · Γ = 8, D1 contains four ofthe Pi’s and D2 passes through the other two, say P1, P2. This implies thatfor the strict transform of D2 we have: D2 ≡ H1 + 3H2 − 2E1 − 2E2, whenceD2 · C = 8, a contradiction. �

We write now G = G1 + G2 + G3 as a sum of its connected components,and accordingly G′ = G′

1 +G′2 +G′

3.

Lemma 2.11. The bidegree of G′j (j ∈ {1, 2, 3}) is (1, 1). Up to renumbering

P1, . . . , P6 we have

G′1 ∩G′

2 = {P1, P2}, G′1 ∩G′

3 = {P3, P4} and G′2 ∩G′

3 = {P5, P6}.

More precisely,

G1 ∈ |H1 +H2 − E1 − E2 − E3 − E4|,G2 ∈ |H1 +H2 − E1 − E2 − E5 − E6|,G3 ∈ |H1 +H2 − E3 − E4 − E5 − E6|.

Proof. Assume for instance that G′1 has bidegree (1, 0). Then there is a subset

I ⊂ {1, . . . , 6} such that G1 = H1−∑i∈I Ei. Since G1 ·C = 0, it follows that

|I| = 2. But then G1 · (G − G1) = 1, contradicting the fact that G1 is aconnected component of G.

Let (aj , bj) be the bidegree of Gj : then aj, bj ≥ 1 since a reduced divisor ofbidegree (m, 0) is not connected for m ≥ 2. Since

∑aj =

∑bj = 3, it follows

that aj = bj = 1.Writing now Gj ≡ H1 +H2 −

∑6i=1 μ(j, i)Ei we obtain

3∑

j=1

μ(j, i) = 2,6∑

i=1

μ(j, i) = 4,6∑

i=1

μ(k, i)μ(j, i) = 2


since Gj · C = 0 and Gk · Gj = 0. We get the second claim of the lemmaprovided that we show: μ(j, i) = 1, ∀i, j.

The first formula shows that if μ(j, i) ≥ 2, then μ(j, i) = 2 and μ(h, i) = 0for h �= j. Hence the second formula shows that

∑

h,k �=j

6∑

i=1

μ(j, i)(μ(h, i) + μ(k, i)) ≤ 2,

contradicting the third formula. �

In the remaining part of the section we will show that each G′i consists of

the union of a vertical and a horizontal line in P1 × P

1.Since OC(KC + η) ∼= OC(H1) and OC(KC − η) ∼= OC(H2) we get:

OC(2H2 −H1) ∼= OC(KC) ∼= OC(2H1 + 2H2 −6∑

i=1

Ei),

whence the exact sequence

0→ OS(−H1 − 4H2 +6∑

i=1

Ei)→ OS(3H1 −6∑

i=1

Ei)

→ OC(3H1 −6∑

i=1

Ei) ∼= OC → 0. (2)

Proposition 2.12. H1(S,OS(−(H1 + 4H2 −∑6i=1 Ei))) = 0.

Proof. The result follows immediately by Ramanujam’s vanishing theorem,but we can also give an elementary proof using Remark 2.7.

It suffices to show that the linear system |H1 + 4H2 −∑6i=1 Ei| contains

a reduced and connected divisor.Note that G1 + |3H2−E5−E6| ⊂ |H1 +4H2−

∑6i=1 Ei|, and that |3H2−

E5 − E6| contains |H2 − E5 − E6| + |2H2|, if there is a line H2 containingP1, P2, else it contains |H2 − E5|+ |H2 − E6|+ |H2|. Since

G1 ·H2 = G1 · (H2 − E5) = G1 · (H2 − E6) = G1 · (H2 − E5 − E6) = 1,

we have obtained in both cases a reduced and connected divisor.�

Remark 2.13. One can indeed show, using

G2 + |3H2 − E3 − E4| ⊂ |H1 + 4H2 −6∑

i=1

Ei|,


G3 + |3H2 − E1 − E2| ⊂ |H1 + 4H2 −6∑

i=1

Ei|,

that |H1 + 4H2 −∑6i=1 Ei| has no fixed part, and then by Bertini’s theorem,

since (H1 +4H2−∑6

i=1 Ei)2 = 8− 6 = 2 > 0, a general curve in |H1 +4H2−∑6

i=1 Ei| is irreducible.

In view of Proposition 2.12 the above exact sequence (and the one wherethe roles of H1, H2 are exchanged) yields the following:

Corollary 2.14. For j ∈ {1, 2} there is exactly one divisor Nj ∈ |3Hj −∑6i=1 Ei|.

By the uniqueness of G, we see that G = N1 +N2. Denote by N ′j the curve

in P1 × P

1 whose total transform is Nj +∑6

i=1 Ei.We have just seen that G is the strict transform of three vertical and three

horizontal lines in P1 × P

1. Hence each connected component Gj splits intothe strict transform of a vertical and a horizontal line. Since G is reduced, thelines are distinct (and there are no infinitely near points).

We can choose coordinates in P1× P

1 such that G′1 = ({∞}× P

1)∪ (P1 ×{∞}), G′

2 = ({0} × P1) ∪ (P1 × {0}), and G′

3 = ({1} × P1) ∪ (P1 × {1}).

Remark 2.15. The points P1, . . . , P6 are then the points of the set S previ-ously defined.

Conversely, consider in P1 × P

1 the set

S := {P1, . . . , P6} = ({∞, 0, 1} × {∞, 0, 1}) \ {(∞,∞), (0, 0), (1, 1)}.

Let π : S → P1 × P

1 be the blowup of the points P1, . . . , P6 and suppose(denoting the exceptional divisor over Pi by Ei) that C ∈ |4H1+4H2−

∑2Ei|

is a smooth curve. Then C has genus 3, OC(3H1) ∼= OC(∑

Ei) ∼= OC(3H2).Setting OC(η) := OC(H2 −H1), we obtain therefore 3η ≡ 0.

It remains to show that OC(η) is not isomorphic to OC .

Lemma 2.16. η is not trivial.

Proof. Assume η ≡ 0. Then OC(H1) ∼= OC(H2) and, since Γ has bidegree(4, 4), we argue as in the proof of Proposition 2.3 that h0(OC(Hi)) ≥ 3,whence OC(Hi) ∼= OC(KC).

The same argument shows that the two projections of Γ to P1 yield two

different pencils in the canonical system. It follows that the canonical mapof C factors as the composition of C → Γ ⊂ P

1 × P1 with the rational map

ψ : P1 × P

1 �� P2 which blows up one point and contracts the vertical and

horizontal line through it. Since Γ has six singular points, the canonical mapsends C birationally onto a singular quartic curve in P

2, contradiction. �


3 Rationality of the moduli spaces

In this section we will use the geometric description of pairs (C, η), where Cis a genus 3 curve and η a nontrivial 3-torsion divisor class, and study thebirational structure of their moduli space.

More precisely, we shall prove the following:

Theorem 3.1.1) The moduli space

M3,η := {(C, η) : C a general curve of genus 3, η ∈ Pic0(C)3 \ {0}}

is rational.2) The moduli space

M3,〈η〉 := {(C, 〈η〉) : C a general curve of genus 3, 〈η〉 ∼= Z/3Z ⊂ Pic0(C)}

is rational.

Remark 3.2. By the result of the previous section, and since any automor-phism of P

1×P1 which sends the set S to itself belongs to the group S3×Z/2Z,

it follows immediately that, if we set

V (4, 4,−S) := H0(OP1×P1(4, 4)(−2∑

i�=j,i,j∈{∞,0,1}Pij)),

then M3,η is birational to P(V (4, 4,−S))/S3, while M3,〈η〉 is birational toP(V (4, 4,−S))/(S3×Z/2Z), where the generator σ of Z/2Z acts by coordinateexchange on P

1 × P1, whence on V (4, 4,−S).

In order to prove the above theorem we will explicitly calculate the respec-tive subfields of invariants of the function field of P(V (4, 4,−S)) and show thatthey are generated by purely transcendental elements.

Consider the following polynomials of V := V (4, 4,−S), which are invari-ant under the action of Z/2Z:

f11(x, y) := x20x

21y

20y

21 ,

f∞∞(x, y) := x21(x1 − x0)2y2

0(y1 − y0)2,

f00(x, y) := x20(x1 − x0)2y2

0(y1 − y0)2.

Let ev : V→⊕

i=0,1,∞ k(i,i) =: W be the evaluation map at the three standarddiagonal points, i.e., ev(f) := (f(0, 0), f(1, 1), f(∞,∞)).

Since fii(j, j) = δi,j , we can decompose V ∼= U ⊕W, where U := ker(ev)and W is the subspace generated by the three above polynomials, which iseasily shown to be an invariant subspace using the following formulae (∗):


• (1, 3) exchanges x0 with x1, multiplies x1 − x0 by −1,

• (1, 2) exchanges x1 − x0 with x1, multiplies x0 by −1,

• (2, 3) exchanges x0 − x1 with x0, multiplies x1 by −1.

In fact, ‘the permutation’ representation W of the symmetric group splits(in characteristic �= 3) as the direct sum of the trivial representation (gen-erated by e1 + e2 + e3) and the standard representation, generated byx0 := e1 − e2, x1 := −e2 + e3, which is isomorphic to the representationon V (1) := H0(OP1(1)).

Note that U = x0x1(x1 − x0)y0y1(y0 − y1)H0(P1 × P1,OP1×P1(1, 1)).

We write

V (1, 1) := H0(P1 × P1,OP1×P1(1, 1)) = V (1)⊗ V (1),

where V (1) := H0(P1,OP1(1)), is as above the standard representation of S3.Now V (1)⊗V (1) splits, in characteristic �= 2, 3, as a sum of irreducible rep-

resentations I⊕A⊕W , where the three factors are the trivial, the alternatingand the standard representation of S3.

Explicitly, V (1) ⊗ V (1) ∼= ∧2(V (1)) ⊕ Sym2(V (1)), and Sym2(V (1)) isisomorphic to W, since it has the following basis: x0y0, x1y1, (x1−x0)(y1−y0).We observe for further use that Z/2Z acts as the identity on Sym2(V (1)),while it acts on ∧2(V (1)), spanned by x1y0 − x0y1 via multiplication by −1.

We have thus seen

Lemma 3.3. If char(k) �= 2, 3, then the S3-module V splits as a sum ofirreducible modules as follows:

V ∼= 2(I⊕W )⊕ A.

Choose now a basis (z1, z2, z3, w1, w2, w3, u) of V, such that the zi’s and thewi’s are respective bases of I⊕W consisting of eigenvectors of σ = (123), andu is a basis element of A. The eigenvalue of zi, wi with respect to σ = (123)is εi−1, u is σ-invariant and (12)(u) = −u.

Note that if (v1, v2, v3) is a basis of I⊕W , such that S3 acts by permutationof the indices, then z1 = v1 +v2 +v3, z2 = v1 + εv2 + ε2v3, z3 = v1 + ε2v2 + εv3,where ε is a primitive third root of unity.

Remark 3.4. Since z1, w1 are S3-invariant, P(V (4, 4,−S))/S3 is birationalto a product of the affine line with Spec(k[z2, z3, w2, w3, u]S3), and thereforeit suffices to compute k[z2, z3, w2, w3, u]S3 .

Part 1 of the theorem follows now from the following

Proposition 3.5. Let T := z2z3, S := z32, A1 := z2w3 + z3w2, A2 := z2w3 −

z3w2. Then


k(z2, z3, w2, w3, u)S3 ⊃ K := k(A1, T, S +T 3

S, u(S − T 3

S), A2(S −

T 3

S)),

and [k(z2, z3, w2, w3, u) : K] = 6, hence k(z2, z3, w2, w3, u)S3 = K.

Proof. We first calculate the invariants under the action of σ = (123), i.e.,k(z2, z3, w2, w3, u)σ. Note that u, z2z3, z2w3, w2w3,z3

2 are σ-invariant, and[k(z2, z3, w2, w3, u) : k(u, z2z3, z2w3, w2w3, z

32)] = 3. In particular,

k(z2, z3, w2, w3, u)σ = k(u, z2z3, z2w3, w2w3, z32) =: L.

Now, we calculate Lτ , with τ = (12). Observe that L = k(T,A1, A2, S, u).Since τ(z2) = εz3, τ(z3) = ε2z2 (and similarly for w2, w3), we see that τ(A1) =A1 and τ(T ) = T . On the other hand, τ(u) = −u, τ(A2) = −A2, τ(S) = T 3

S .

Claim.Lτ = k(A1, T, S + T 3

S , u(S − T 3

S ), A2(S − T 3

S )) =: E.

Proof of the Claim. Obviously A1,T ,S+ T 3

S ,u(S− T 3

S ), A2(S− T 3

S ) are invariantunder τ , whence E ⊂ Lτ . Since L = E(S), using the equation B ·S = S2 +T 3

for B := S + T 3

S , we get that [E(S) : E] ≤ 2.

This proves the claim and the proposition. �

It remains to show the second part of the theorem. We denote by τ ′ theinvolution on k(z1, z2, z3, w1, w2, w3, u) induced by the involution (x, y) �→(y, x) on P

1 × P1. It suffices to prove the following

Proposition 3.6. Eτ′= k(A1, T, S + T 3

S , (u(S − T 3

S ))2, A2(S − T 3

S )).

Proof. Since [E : k(A1, T, S + T 3

S , (u(S − T 3

S ))2, A2(S − T 3

S ))] ≤ 2, it sufficesto show that the five generators A1, T ,S + T 3

S ,(u(S − T 3

S ))2, A2(S − T 3

S ) areτ ′-invariant. This will now be proven in Lemma 3.7. �

Lemma 3.7. τ ′ acts as the identity on (z1, z2, z3, w1, w2, w3) and sends u �→−u.

Proof. We note first that τ ′ acts trivially on the subspace W generated by thepolynomials fii.

Since U = x0x1(x1−x0)y0y1(y1−y0)V (1, 1) and x0x1(x1−x0)y0y1(y1−y0)is invariant under exchanging x and y, it suffices to recall that the action ofτ ′ on V (1, 1) = V (1) ⊗ V (1) is the identity on the subspace Sym2(V (1)),while the action on the alternating S3-submodule A sends the generator u to−u. �


3.1 Char(k) = 3

In order to prove Theorem 3.1 if the characteristic of k is equal to 3, wedescribe the S3-module V as follows:

V ∼= 2W⊕ A,

where W is the (three-dimensional) permutation representation of S3.Let now z1, z2, z3, w1, w2, w3, u be a basis of V such that the action of S3

permutes z1, z2, z3 (resp. w1, w2, w3), and (123) : u �→ u, (12)u �→ −u. Thenwe have:

Proposition 3.8. The S3-invariant subfield k(V)S3 of k(V) is rational.More precisely, the seven S3-invariant functions

σ1 = z1 + z2 + z3,

σ2 = z1z2 + z1z3 + z2z3,

σ3 = z1z2z3,

σ4 = z1w1 + z2w2 + z3w3,

σ5 = w1z2z3 + w2z1z3 + w3z1z2,

σ6 = w1(z2 + z3) + w2(z1 + z3) + w3(z1 + z2),

σ7 = u(z1(w2 − w3) + z2(w3 − w1) + z3(w1 − w2))

form a basis of the purely transcendental extension over k.

Proof. σ1, . . . , σ7 determine a morphism ψ : V → A7k. We will show that ψ

induces a birational map ψ : V/S3 → A7k, i.e., for a Zariski open set of V we

have: ψ(x) = ψ(x′) if and only if there is a τ ∈ S3 such that x = τ(x′). By[Cat, Lemma 2.2] we can assume (after acting on x with a suitable τ ∈ S3)that xi = x′i for 1 ≤ i ≤ 6, and we know that (setting u := x7, u′ := x′7)

u(x1(x5 − x6) + x2(x6 − x4) + x3(x4 − x5)) =u′(x1(x5 − x6) + x2(x6 − x4) + x3(x4 − x5)).

Therefore, if B(x1, . . . , x6) := x1(x5 − x6) + x2(x6 − x4) + x3(x4 − x5) �= 0,this implies that u = u′. �

Therefore, we have shown part 1 of Theorem 3.1.We denote again by τ ′ the involution on k(z1, z2, z3, w1, w2, w3, u) induced

by the involution (x, y) �→ (y, x) on P1 × P

1. In order to prove part 2 of The-orem 3.1, it suffices to observe that σ1, . . . , σ6, σ

27 are invariant under τ ′ and

[k(σ1, . . . , σ7) : k(σ1, . . . , σ27)] ≤ 2, whence (k(V)S3 )(Z/2Z) = k(σ1, . . . , σ

27).

This proves Theorem 3.1.


3.2 Char(k) = 2

Let k be an algebraically closed field of characteristic 2. Then we can describethe S3-module V as follows:

V ∼= W⊕ V (1, 1),

where W is the (three-dimensional) permutation representation of S3. Wedenote a basis of W by z1, z2, z3. As in the beginning of the chapter, V (1, 1) =H0(P1 × P

1,OP1×P1(1, 1)). We choose the following basis of V (1, 1): w1 :=x1y1, w2 := (x0 + x1)(y0 + y1), w3 := x0y0, w := x0y1. Then S3 acts onw1, w2, w3 by permutation of the indices and

(1, 2) : w �→ w + w3,

(1, 2, 3) : w �→ w + w2 + w3.

Let ε ∈ k be a nontrivial third root of unity. Then Theorem 3.1 (in character-istic 2) follows from the following result:

Proposition 3.9. Let k be an algebraically closed field of characteristic 2. Letσ1, . . . , σ6 be as defined in (3.6) and set

v := (w + w2)(w1 + εw2 + ε2w3) + (w + w1 + w3)(w1 + ε2w2 + εw3),

t := (w + w2)(w + w1 + w3).

Then

1) k(z1, z2, z3, w1, w2, w3, w)S3 = k(σ1, . . . , σ6, v);

2) k(z1, z2, z3, w1, w2, w3, w)S3×Z/2Z = k(σ1, . . . , σ6, t).

In particular, the respective invariant subfields of k(V) are generated bypurely transcendental elements, and this proves Theorem 3.1.

Proof (of Proposition 3.9). 2) We observe that Z/2Z (xi �→ yi) acts triv-ially on z1, z2, z3, w1, w2, w3 and maps w to w + w1 + w2 + w3. It is noweasy to see that t is invariant under the action of S3 × Z/2Z. There-fore k(σ1, . . . , σ6, t) ⊂ K := k(z1, z2, z3, w1, w2, w3, w)S3×Z/2Z. By [Cat,Lemma 2.8], [k(z1, z2, z3, w1, w2, w3, t) : k(σ1, . . . , σ6, t)] = 6, and obvi-ously, [k(z1, z2, z3, w1, w2, w3, w) : k(z1, z2, z3, w1, w2, w3, t)] = 2. Therefore[k(z1, z2, z3, w1, w2, w3, w) : k(σ1, . . . , σ6, t)] = 12, whenceK = k(σ1, . . . , σ6, t).

1) Note that for W2 := w1 + εw2 + ε2w3, W3 := w1 + ε2w2 + ε3w3,we have: W 3

2 and W 33 are invariant under (1, 2, 3) and are exchanged un-

der (1, 2). Therefore v is invariant under the action of S3 and we haveseen that k(σ1, . . . , σ6, v) ⊂ L := k(z1, z2, z3, w1, w2, w3, w)S3 , in particular[k(z1, z2, z3, w1, w2, w3, w) : k(σ1, . . . , σ6, v)] ≥ 6. On the other hand, notethat k(z1, z2, z3, w1, w2, w3, w) = k(z1, z2, z3, w1, w2, w3, v) (since v is linear inw) and again, by [Cat, Lemma 2.8], [k(zi, wi, v) : k(σ1, . . . , σ6, v)] = 6. Thisimplies that L = k(σ1, . . . , σ6, v). �


References

[B-C04] Bauer, I. and Catanese, F., Symmetry and variation of Hodge struc-tures, Asian J. Math., vol.8, no.2, 363–390, (2004).

[B-K85] Bogomolov, F. A. and Katsylo, P. I., Rationality of some quotientvarieties, Mat. Sb. (N.S.), vol. 126 (168), no. 4, 584–589, (1985).

[Cat] Catanese, F., On the rationality of certain moduli spaces related tocurves of genus 4, Algebraic geometry (Ann Arbor, Mich., 1981), 30–50,Lecture Notes in Math., 1008, Springer, Berlin (1983).

The Rationality of the Moduli Space of Curvesof Genus 3 after P. Katsylo

Christian Böhning

Mathematisches InstitutBunsenstrasse 3-537073 Göttingen, [email protected]

Summary. This article is a survey of P. Katsylo’s proof that the moduli spaceM3 of smooth projective complex curves of genus 3 is rational. We hope to makethe argument more comprehensible and transparent by emphasizing the underlyinggeometry in the proof and its key structural features.

Key words: Rationality, moduli spaces of curves

2000 Mathematics Subject Classification codes: 14E08, 14H10, 14H45

1 Introduction

The question whether or not M3 is a rational variety had been open for along time until an affirmative answer was finally given by P. Katsylo in 1996.There is a well-known transition in the behavior of the moduli spaces Mg ofsmooth projective complex curves of genus g from unirational for small g togeneral type for larger values of g; the moral reason that M3 should have agood chance to be rational is that it is birational to a quotient of a projectivespace by a connected linear algebraic group. No variety of this form has beenproved to be irrational up to now. More precisely, M3 is birational to themoduli space of plane quartic curves for PGL3 C-equivalence. All the modulispaces C(d) of plane curves of given degree d are conjectured to be rational(see [Dol2, p.162]; in fact, there it is conjectured that all the moduli spaces ofhypersurfaces of given degree d in P

n for the PGLn+1 C-action are rational).There are heuristic reasons that the spaces C(d) should be rational at least

for all large enough values for d. It should not be completely out of reach toprove this rigorously and we hope to return to this problem in the future. Inany case, one might guess that irregular behavior of C(d) is most likely to befound for small values of d, and showing rationality for C(4) turned out to beexceptionally hard.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_2, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

18 C. Böhning

Katsylo’s proof is long and computational, and, due to the importance ofthe result, it seems desirable to give a more accessible and geometric treatmentof the argument.

This paper is divided into two main sections (Sections 2 and 3) whichare further divided into subsections. Section 2 treats roughly the contents ofKatsylo’s first paper [Kat1] and Section 3 deals with his second paper [Kat2].

Acknowledgment. I would like to thank Professor Yuri Tschinkel for propos-ing the project and many useful discussions. Moreover, I am especially gratefulto Professor Fedor Bogomolov with whom I discussed parts of the project andwho provided a wealth of helpful ideas.

2 A remarkable (SL3 C, SO3 C)-section

2.1 (G, H)-sections and covariants

A general, i.e., nonhyperelliptic, smooth projective curve C of genus 3 is real-ized as a smooth plane quartic curve via the canonical embedding, whence M3

is birational to the orbit space C(4) := P(H0(P2,O(4)))/SL3 C. We remarkthat whenever one has an affine algebraic group G acting on an irreduciblevariety X , then, according to a result of Rosenlicht, there exists a nonemptyinvariant open subset X0 ⊂ X such that there is a geometric quotient for theaction of G on X0 (cf. [Po-Vi, Theorem 4.4]). In the following we denote byX/G any birational model of this quotient, i.e., any model of the field C(X)G

of invariant rational functions.The number of methods to prove rationality of quotients of projective

spaces by connected reductive groups is quite limited (cf. [Dol1] for an excel-lent survey). The only approach which our problem is immediately amenableto seems to be the method of (G,H)-sections. (There are two other pointsof view I know of: The first is based on the remark that if we have a non-singular plane quartic curve C, the double cover of P

2 branched along C is aDel Pezzo surface of degree 2, and conversely, given a Del Pezzo surface S ofdegree 2, then | −KS| is a regular map which exhibits S as a double cover ofP

2 branched along a plane quartic C; this sets up a birational isomorphismbetween M3 and DP(2), the moduli space of Del Pezzo surfaces of degree 2.We can obtain such an S by blowing up 7 points in P

2, and one can prove thatDP(2) is birational to the quotient of an open subset of P 7

2 := (P2)7/PGL3 C,the configuration space of 7 points in P

2 (which is visibly rational), moduloan action of the Weyl group W (E7) of the root system of type E7 by Cremonatransformations (note that W (E7) coincides with the permutation group ofthe (−1)-curves on S that preserves the incidence relations between them).This group is a rather large finite group, in fact, it has order 210 · 34 · 5 · 7.This approach does not seem to have led to anything definite in the directionof proving rationality of M3 by now, but see [D-O] for more information.

Rationality of M3 19

The second alternative, pointed out by I. Dolgachev, is to remark thatM3 is birational to Mev

3 , the moduli space of genus 3 curves together with aneven theta-characteristic; this is the content of the classical theorem due toG. Scorza. The latter space is birational to the space of nets of quadrics inP

3 modulo the action of SL4 C, i.e., Grass(3, Sym2 (C4)∨)/SL4 C. See [Dol3,6.4.2], for more on this. Compare also [Kat0], where the rationality of therelated space Grass(3, Sym2 (C5)∨)/SL5 C is proven; this proof, however, can-not be readily adapted to our situation, the difficulty seems to come down tothat 4, in contrast to 5, is even.

Definition 2.1.1. Let X be an irreducible variety with an action of a linearalgebraic group G, H < G a subgroup. An irreducible subvariety Y ⊂ X iscalled a (G,H)-section of the action of G on X if

(1)G · Y = X;(2)H · Y ⊂ Y ;(3) g ∈ G, gY ∩ Y �= ∅ =⇒ g ∈ H.

In this situation H is the normalizer NG(Y ) := {g ∈ G | gY ⊂ Y } of Y in G.The following proposition collects some properties of (G,H)-sections.

Proposition 2.1.2. (1)The field C(X)G is isomorphic to the field C(Y )H viarestriction of functions to Y .

(2) Let Z and X be G-varieties, f : Z → X a dominant G-morphism, Y a(G,H)-section of X, and Y ′ an irreducible component of f−1(Y ) that isH-invariant and dominates Y . Then Y ′ is a (G,H)-section of Z.

Part (2) of the proposition suggests that, to simplify our problem of provingrationality of C(4), we should look at covariants Sym4 (C3)∨ → Sym2 (C3)∨

of low degree (C3 is the standard representation of SL3 C). The highest weighttheory of Cartan–Killing allows us to decompose Symi(Sym4 (C3)∨), i ∈ N,into irreducible subrepresentations (this is best done by a computer algebrasystem, e.g., Magma) and pick the smallest i such that Sym2 (C3)∨ occurs asan irreducible summand. This turns out to be 5 and Sym2 (C3)∨ occurs withmultiplicity 2.

For nonnegative integers a, b we denote by V (a, b) the irreducible SL3 C-module whose highest weight has numerical labels a, b.

Let us now describe the two resulting independent covariants

α1, α2 : V (0, 4)→ V (0, 2)

of order 2 and degree 5 geometrically. We follow a classical geometric methodof Clebsch to pass from invariants of binary forms to contravariants of ternaryforms (see [G-Y, §215]). The covariants α1, α2 are described in Salmon’streatise [Sal, p. 261, and p. 259], cf. also [Dix, pp. 280–282]. We start byrecalling the structure of the ring of SL2 C-invariants of binary quartics ([Muk,Section 1.3], [Po-Vi, Section 0.12]).

20 C. Böhning

2.2 Binary quartics

Let

f4 = ξ0x40 + 4ξ1x3

0x1 + 6ξ2x20x

21 + 4ξ3x0x

31 + ξ4x

41 (1)

be a general binary quartic form. The invariant algebra R = C[ξ0, . . . , ξ4]SL2 C

is freely generated by two homogeneous invariants g2 and g3 (where subscriptsindicate degrees):

g2(ξ) = det(ξ0 ξ2ξ2 ξ4

)− 4 det

(ξ1 ξ2ξ2 ξ3

), (2)

g3(ξ) = det

⎛

⎝ξ0 ξ1 ξ2ξ1 ξ2 ξ3ξ2 ξ3 ξ4

⎞

⎠ . (3)

If we identify f4 with its zeros z1, . . . , z4 ∈ P1 = C ∪ {∞} and write

λ =(z1 − z3)(z2 − z4)(z1 − z4)(z2 − z3)

for the cross-ratio, then

g3 = 0 ⇐⇒ λ = −1, 2, or12,

g2 = 0 ⇐⇒ λ = −ω or − ω2 with ω = e2πi3 ,

the first case being commonly referred to as harmonic cross-ratio, the secondas equi-anharmonic cross-ratio (see [Cle, p. 171]; the terminology varies a lotamong different authors, however).

Clebsch’s construction is as follows: Let x, y, z be coordinates in P2, and

let u, v, w be coordinates in the dual projective plane (P2)∨. Let

ϕ = ax4 + 4bx3y + . . .

be a general ternary quartic. We consider those lines in P2 such that their

intersection with the associated quartic curve Cϕ is a set of points whose cross-ratio is harmonic resp. equi-anharmonic. Writing a line as ux + vy + wz = 0and substituting in (2) resp. (3), we see that in the equi-anharmonic case weget a quartic in (P2)∨, and in the harmonic case a sextic. More precisely, thisgives us two SL3 C-equivariant polynomial maps

σ : V (0, 4)→ V (0, 4)∨ , (4)ψ : V (0, 4)→ V (0, 6)∨ , (5)

and σ is homogeneous of degree 2 in the coefficients of ϕ whereas ψ is ho-mogeneous of degree 3 in the coefficients of ϕ (we say σ is a contravariant of

Rationality of M3 21

degree 2 on V (0, 4) with values in V (0, 4), and analogously for ψ). Finally wehave the Hessian covariant of ϕ:

Hess : V (0, 4)→ V (0, 6) (6)

which associates to ϕ the determinant of the matrix of second partial deriva-tives of ϕ. It is of degree 3 in the coefficients of ϕ.

We will now cook up α1, α2 from ϕ, σ, ψ, Hess: Let ϕ operate on ψ; by thiswe mean that if ϕ = ax4 + 4bx3y + . . . , then we act on ψ by the differentialoperator

a∂4

∂u4+ 4b

∂4

∂u3∂v+ . . .

(i.e., we replace a coordinate by partial differentiation with respect to the dualcoordinate). In this way we get a contravariant ρ of degree 4 on V (0, 4) withvalues in V (0, 2). If we operate with ρ on ϕ, we get α1. We obtain α2 if weoperate with σ on Hess.

This is a geometric way to describe α1, α2. For every c = [c1 : c2] ∈ P1 we

get in this way a rational map

fc = c1α1 + c2α2 : P(V (0, 4)) �� P(V (0, 2)) . (7)

For the special quartics

ϕ = ax4 + by4 + cz4 + 6fy2z2 + 6gz2x2 + 6hx2y2 (8)

the quantities α1 and α2 were calculated by Salmon in [Sal, p. 257 ff]. Wereproduce the results here for the reader’s convenience. Put

L := abc , P := af2 + bg2 + ch2 , (9)R := fgh. ;

Then

α1 = (3L+ 9P + 10R)(afx2 + bgy2 + chz2) (10)

+(10L+ 2P + 4R)(ghx2 + hfy2 + fgz2)

−12(a2f3x2 + b2g3y2 + c2h3z2) ;

α2 = (L+ 3P + 30R)(afx2 + bgy2 + chz2) (11)

+(10L− 6P − 12R)(ghx2 + hfy2 + fgz2)

−4(a2f3x2 + b2g3y2 + c2h3z2) .

Note that the covariant conic − 120 (α1 − 3α2) looks a little simpler.

Let us see explicitly, using (8)–(11), that fc is dominant for every c ∈ P1;

for a = b = c = f = g = h = 1 we get α1 = 48(x2 + y2 + z2), α2 =16(x2 + y2 + z2), so the image of ϕ under fc in this case is a nonsingular

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