Table Of ContentProgress in Mathematics
Volume282
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Cohomological
and Geometric Approaches
to Rationality Problems
New Perspectives
Fedor Bogomolov
Yuri Tschinkel
Editors
Birkha¨user
Boston (cid:129) Basel (cid:129) Berlin
FedorBogomolov YuriTschinkel
DepartmentofMathematics DepartmentofMathematics
NewYorkUniversity NewYorkUniversity
CourantInstitute CourantInstitute
ofMathematicalsciences ofMathematicalsciences
NewYork,NY10012 NewYork,NY10012
U.S.A U.S.A
[email protected] [email protected]
ISBN978-0-8176-4933-3 e-ISBN978-0-8176-4934-0
DOI10.1007/978-0-8176-4934-0
LibraryofCongressControlNumber:2009939069
MathematicsSubjectClassification(2000):11R32,12F12,13A50,14D20,14E05,14E08,14F20,14G05,
14G15,14H10,14H45,14H60,14J32,14J35,14L30
(cid:2)c Birkha¨userBoston,apartofSpringerScience+BusinessMedia,LLC2010
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Contents
Preface............................................................... vii
The Rationality of Certain Moduli Spaces of Curves of
Genus 333
Ingrid Bauer and Fabrizio Catanese................................... 1
The Rationality of the Moduli Space of Curves of Genus333 after
P. Katsylo
Christian Bo¨hning..................................................... 17
Unramified Cohomology of Finite Groups of Lie Type
Fedor Bogomolov, Tihomir Petrov and Yuri Tschinkel................. 55
Sextic Double Solids
Ivan Cheltsov and Jihun Park......................................... 75
Moduli Stacks of Vector Bundles on Curves and the King–
Schofield Rationality Proof
Norbert Hoffmann..................................................... 133
Noether’s Problem for Some ppp-Groups
Shou-Jen Hu and Ming-chang Kang................................... 149
Generalized Homological Mirror Symmetry and
Rationality Questions
Ludmil Katzarkov..................................................... 163
The Bogomolov Multiplier of Finite Simple Groups
Boris Kunyavski˘ı...................................................... 209
vi Contents
Derived Categories of Cubic Fourfolds
Alexander Kuznetsov.................................................. 219
Fields of Invariants of Finite Linear Groups
Yuri G. Prokhorov..................................................... 245
The Rationality Problem and Birational Rigidity
Aleksandr V. Pukhlikov................................................ 275
Preface
The rationality problem links algebra and geometry. On the level of fields it
can be formulated as follows: Let K be a field of finite transcendence degree
over a ground field k, i.e., K is the field of rational functions on an algebraic
variety defined over k. Decide whether or not K is a purely transcendental
extension of k. It became apparent that, for k = C, the geometry of the
corresponding variety is tightly linked with rationality of the field K.
The difficulty of the rationality problem depends on the transcendence
degreeofK overk or,geometrically,onthedimensionofthevariety.Amajor
success of the 19th and the first half of the 20th century algebraic geometry
wasacompletesolutionofthe rationalityproblemindimensions oneandtwo
over algebraically closed ground fields of characteristic zero. In these cases it
suffices to consider finite, easily computable geometric invariants. For such
fields in dimension two, the presence of sufficiently flexible rational curves
suffices for the rationality of the field. This geometric property (rationalcon-
nectedness) defines a class of algebraic varieties (and fields) which in higher
dimensions is substantially larger than the class of rational varieties: it is
known that most of rationally connected varieties are not rational already in
dimension three. It is still an open question whether or not they are unira-
tional, i.e., the corresponding function fields are proper subfields of rational
fields.
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 of
linearactionsofgroupsoncomplexlinearspaces—manybeautifulexplicitcon-
structions can be traced back to the works of Sylvester, Cayley, and Gordan.
Emmy Noether,who was a student of PaulGordan,one of the experts in the
theory of explicit invariants,formulateda generalquestion:are all such fields
ofinvariantsrational?Thefirstcounterexamples,overnonclosedgroundfields,
were constructed by Swan, and over the complex numbers, by Saltman. This
made it even more interesting to determine all rational fields of invariants.
viii Preface
ThearticlebyProkhorovreviewsknownresultsincaseoffour-dimensional
complex representations of finite groups. It is shown that almost all of the
arisingquotients are rational.The paper by Hu and Kangcontains a proofof
rationalityforquotientsbysmallp-groups.Kunyavski˘ıprovestrivialityofthe
first cohomologicalobstruction to rationality, for all finite simple groups and
quasi-simple groups of Lie type. He also shows the existence of exceptions—
there are quasi-simple groups which are central extensions of one particular
simple group of Lie type with a nonvanishing obstruction. Remarkably, these
areexactlytheextensionswhicharenotofLietypethemselves.Thepaperby
Bogomolov, Petrov, and Tschinkel is devoted to more general cohomological
invariantsofquasi-simplegroupsofLie type.Itis shownthatallobstructions
to rationality coming from nonramified cohomology vanish for a large class
of quasi-simple groups of Lie type, at least when the groupof coefficients has
ordercoprimeto the characteristicofthe fieldofdefinition ofthe group.Two
paperstreatrationalityofthemodulispaceofcurvesofgenusthree.Thiswas
established 15 years ago by Katsylo but the proof was computationally very
involved and difficult to follow. C. Bo¨hning wrote a detailed and transparent
version of this proof. Bauer and Catanese prove rationality of the related
modulispaceofcurvesofgenusthreewithafixednontrivial3-torsionpointin
theJacobian.Theyidentifythismodulispacewiththemodulispaceofcurves
of bidegree (4,4) ⊂ P1×P1, with simple singularities at the presecribed set
of six points. Hoffmann’s paper is devoted to rationality of moduli spaces of
vector 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 a
projective curve having a fixed determinant line bundle, with coprime rank
and degree of the determinant.
Rationalfields havemany automorphismsandthe absenceofthese points
to nonrationality. Pukhlikov surveys nonrationality results obtained via the
studyofrigiditypropertiesoftheprojectivemodelsofthefield.Firstexamples
ofrigidrationallyconnectedvarietieswerediscoveredbyManinandIskovskikh
in the seventies. Cheltsov and Park consider an arithmetic aspect of ratio-
nality: a classical diophantine problem concerns the description of rational
solutions of a system of polynomial equations, i.e., rational points on alge-
braic varieties. If the variety is rational or unirational, then there are many
suchpoints, atleastovera finite extensionof the groundfield. This potential
density of rational points holds for all Fano varieties of dimension ≤ 3 with
an exception of one class: double covers of the three-dimensional projective
space ramified over a surface of degree 6. Cheltsov–Park show that potential
density of rational points holds if the ramification surface has at least one
singular point. They also show that these varieties are superrigid.
Katzarkovintroducesacompletelynovelapproachtotherationalityprob-
lem, based on ideas of Mirror Symmetry. The Mirror Symmetry conjecture
is still open but the ideas suggested by physics already highlight new inter-
esting phenomena related to rationality.Applications of derivedcategoriesto
rationality questions are discussed in the paper by Kuznetsov.He studies the
Preface ix
derivedcategoryof coherentsheavesof four-dimensionalcubic hypersurfaces.
It is expected that a generic cubic fourfold is nonrational;this remains one of
the challenging problems in the field.
Needless to say, the book does not cover the whole spectrum of modern
approachesto rationality.Nevertheless, we hope that it provides a glimpse of
the variety of related concrete problems, new methods and results.
Fedor Bogomolov
Yuri Tschinkel
April 2009
The Rationality of Certain Moduli Spaces
of Curves of Genus 3
Ingrid Bauer and Fabrizio Catanese
Mathematisches Institut
Universität Bayreuth, NW II
D-95440 Bayreuth,Germany
[email protected], [email protected]
Summary. We prove rationality of the moduli space of pairs of curves of genus
threetogether 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-
tionalstructureofthemodulispaceofpairs(C,η),whereC isageneralcurve
of genus 3 over an algebraically closed field k of arbitrary characteristic and
η ∈Pic0(C) is a nontrivial divisor class of 3-torsionon C.
3
As was observed in [B-C04, Lemma (2.18)], if C is a general curve of
genus 3 and η ∈ Pic0(C) is a nontrivial 3-torsion divisor class, then we
3
have a morphism ϕη := ϕ|KC+η| ×ϕ|KC−η| : C → P1 ×P1, corresponding
to the sum of the linear systems |K +η| and |K −η|, which is birational
C C
onto a curve Γ ⊂ P1 × P1 of bidegree (4,4). Moreover, Γ has exactly six
ordinary double points as singularities, located in the six points of the set
S :={(x,y)|x(cid:5)=y, x,y ∈{0,1,∞}}.
In [B-C04] we only gave an outline of the proof (and there is also a minor
inaccuracy).Thereforewededicatethefirstsectionofthisarticletoadetailed
geometricaldescriptionofsuchpairs(C,η),whereC isageneralcurveofgenus
3 and η ∈Pic0(C) \{0}.
3
The main result of the first section is the following:
Theorem 1.1.Let C be a general (in particular, nonhyperelliptic) curve of
genus 3 over an algebraically closed field k (of arbitrary characteristic) and
η ∈Pic0(C) \{0}.
3
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
