Geometry and Arithmetic

383 Pages·2012·6.03 MB·English

by  Carel Faber

EMS Series of Congress Reports
EMS Congress Reportspublishes volumes originating from conferences or seminars focusing on any field
of pure or applied mathematics.The individual volumes include an introduction into their subject and
review of the contributions in this context.Articles are required to undergo a refereeing process and are
accepted only if they contain a survey or significant results not published elsewhere in the literature.
Previously published:
Trends in Representation Theory of Algebras and Related Topics,Andrzej Skowron´ski (ed.)
K-Theory and Noncommutative Geometry,Guillermo Cortiñaset al. (eds.)
Classification of Algebraic Varieties,Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.)
Surveys in Stochastic Processes,Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.)
Representations of Algebras and Related Topics,Andrzej Skowron´ski and Kunio Yamagata (eds.)
Contributions to Algebraic Geometry.Impanga Lecture Notes,Piotr Pragacz (ed.)
Geometry and
Arithmetic
Carel Faber
Gavril Farkas
Robin de Jong
Editors
Editors:
Carel Faber Gavril Farkas Robin de Jong
Department of Mathematics Institut für Mathematik Mathematical Institute
Royal Institute of Technology Humboldt-Universität zu Berlin University of Leiden
100 44 Stockholm Unter den Linden 6 P.O.Box 9512
Sweden 10099 Berlin 2300 RA Leiden
Germany The Netherlands
E-mail:[email protected]
E-mail:[email protected] E-mail:[email protected]
2010 Mathematics Subject Classification:14;11
Key words:Algebraic geometry,arithmetic geometry
ISBN 978-3-03719-119-4
The Swiss National Library lists this publication in The Swiss Book,the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright.All rights are reserved,whether the whole or part of the material
is concerned,specifically the rights of translation,reprinting,re-use of illustrations,recitation,
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use permission of the copyright owner must be obtained.
© 2012 European Mathematical Society
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Printing and binding:Beltz Bad Langensalza GmbH,Bad Langensalza,Germany
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9 8 7 6 5 4 3 2 1
Contents
Preface.......................................................................vii
Nef divisors on M from GIT
0,n
Valery Alexeev and David Swinarski............................................1
Inoue type manifolds and Inoue surfaces: a connected component
of the moduli space of surfaces with K2 =7, p =0
g
Ingrid Bauer and Fabrizio Catanese...........................................23
Non-rationality of the symmetric sextic Fano threefold
Arnaud Beauville.............................................................57
Brill–Noether loci of stable rank–two vector bundles on a general curve
Ciro Ciliberto and Flaminio Flamini..........................................61
Mordell-Weil groups and Zariski triples
Jos´e Ignacio Cogolludo-Agust´ın and Remke Kloosterman......................75
Approximate computations with modular curves
Jean-Marc Couveignes and Bas Edixhoven....................................91
A remark on a conjecture of Paranjape and Ramanan
Friedrich Eusen and Frank-Olaf Schreyer....................................113
On extensions of the Torelli map
Angela Gibney...............................................................125
The classes of singular moduli in the generalized Jacobian
Benedict H. Gross...........................................................137
The Eisenstein motive for the cohomology of GSp (Z)
2
Gu¨nter Harder...............................................................143
Cohomology of the moduli stack of coherent sheaves on a curve
Jochen Heinloth.............................................................165
New methods for bounding the number of points on curves over finite fields
Everett W. Howe and Kristin E. Lauter......................................173
vi Contents
Wildly ramified actions and surfaces of general type arising
from Artin-Schreier curves
Hiroyuki Ito and Stefan Schr¨oer.............................................213
A note on a supersingular K3 surface in characteristic 2
Toshiyuki Katsura and Shigeyuki Kond¯o.....................................243
The intuitive definition of Du Bois singularities
S´andor J Kov´acs............................................................257
Bundles of rank 2 with small Clifford index on algebraic curves
H. Lange and P. E. Newstead................................................267
Descendents on local curves: Stationary theory
R. Pandharipande and A. Pixton............................................283
A remark on Getzler’s semi-classical approximation
Dan Petersen................................................................309
On the modular curve X (23)
0
Ren´e Schoof.................................................................317
Degree 4 unramified cohomology with finite coefficients
and torsion codimension 3 cycles
Claire Voisin................................................................347
Poincar´e duality and unimodularity
Yuri G. Zarhin..............................................................369
Preface
This volume contains 21 invited papers in algebraic and arithmetic geometry.
Thetreatedtopicsrangeoveravarietyofthemes,includingmodulispacesofcurves
andabelianvarieties,algebraiccycles,vectorbundlesandcoherentsheaves,curves
over finite fields, and algebraic surfaces, among others.
Thevolumeislooselyconnectedtotheconference“GeometryandArithmetic”
that took place on the island of Schiermonnikoog, The Netherlands, in September
2010. Roughly half the papers here were submitted by conference participants;
theothercontributionswereinvitedbyusanddiscusssubjectsthatfitthegeneral
themes of the conference well. Each article has been carefully refereed.
ItisapleasuretodedicatethisvolumetoGerardvanderGeer,ourPh.D.advi-
sor, who continues to inspire us. Gerard’s 60th birthday provided the occasion for
the organization of the conference; the two main themes of this volume and of the
conference were chosen to represent and reflect his broad mathematical interests.
TohonourGerard’sfondnessofandinterestinforeignlanguages,manyauthors
have provided a second abstract in their native language. We hope that most
readers will enjoy this special feature of the volume.
Wetaketheopportunitytothanktheauthorsandrefereesofthearticles,aswell
asthespeakersandparticipantsoftheconference,fortheircontributions. Lastbut
not least, we thank the institutions whose financial support made the conference
possible: the Foundation Compositio Mathematica, the research cluster Geome-
try and Quantum Theory, the research school WONDER, and the Netherlands
Organization for Scientific Research.
July 2012 Carel Faber
Gavril Farkas
Robin de Jong
Nef divisors on M from GIT
0,n
Valery Alexeev and David Swinarski∗
Abstract. We introduce and study the GIT cone of M , which is generated by the
0,n
pullbacksofthenaturalamplelinebundlesontheGITquotients(P1)n//SL(2). Wegive
an explicit formula for these line bundles and prove a number of basic results about the
GIT cone. As one application, we prove unconditionally that the log canonical models
ofM withasymmetricboundarydivisorcoincidewiththemodulispacesofweighted
0,n
curves or with the symmetric GIT quotient, extending the result of Matt Simpson.
Rez(cid:24)me. Vdanno(cid:26)stat~eopredel(cid:31)ets(cid:31)iissleduets(cid:31) “GITkonus"naprostranstve
module(cid:26) stabil~nyh krivyh M , kotory(cid:26) poro(cid:25)den proobrazami estestvennyh
0,n
obil~nyh line(cid:26)nyh rassloeni(cid:26) na faktorah (P1)n//SL(2). Nahod(cid:31)ts(cid:31) (cid:31)vnye for-
mulydl(cid:31)(cid:11)tihline(cid:26)nyhrassloeni(cid:26)idokazyvaets(cid:31)r(cid:31)dfundamental~nyhrezul~ta-
tovoGITkonuse.Kakodnoizprimeneni(cid:26),dokazyvaets(cid:31),bezovs(cid:31)kihdopolnitel~-
nyhuslovi(cid:26),qtologkanoniqeskiemodeliM ssimmetriqnymigranicamisovpa-
0,n
da(cid:24)tsmodul(cid:31)mivzvexennyhstabil~nyhkrivyhilissimmetriqeskimfaktorom,
obobwa(cid:31) rezul~tat M(cid:11)tt~(cid:24) Simpsona.
2010 Mathematics Subject Classification. 14D22, 14H10, 14L24, 14Q05.
Keywords. Moduli spaces of curves, Geometric Invariant Theory, GIT.
1. Introduction
ThemodulispaceofsmoothpointedgenuszerocurvesM hasmanycompactifi-
0,n
cations. AmongthemostimportantoftheseisM ,themodulispaceofDeligne-
0,n
Mumford-Knudsen’s stable curves. There are many beautiful results on these
spaces; for instance, Keel and Kapranov described them as explicit blowups of
(P1)n−3 and Pn−3 ([19, 18]), and Kapranov identifed M with the Chow quo-
0,n
tient of the Grassmannian Gr(2,n) by a torus. But the birational geometry of
these spaces is still not fully understood, and it is known to be very complicated.
For example, Faber computed that the nef cone of M has 3190 extremal rays.
0,6
Another family of compactifications of M is provided by the GIT quotients
0,n
(P1)n// SL(2), where the vector x specifies the linearization O(x ,...,x ). Let
x 1 n
L denotethedistinguishedpolarizationontheGITquotientdescendingfromthe
x
linearization.
By[17],therearebirationalmorphismsπ :M →(P1)n// SL(2),andhence
x 0,n x
the pullbacks L :=π∗L of the distinguished polarizations on the GIT quotients
x x x
∗We would like to thank Angela Gibney and David Smyth for helpful discussions regarding
this work. We are also grateful to Boris Alexeev, who wrote a C++ program for finding the
0-cellsoftheGITcomplex.
2 Valery Alexeev and David Swinarski
are big and nef, since π is birational and L is ample. In this paper we consider
x x
thesubconegeneratedbytheL insidethenefconeofM . WecallthistheGIT
x 0,n
cone and denote it GC.
Our first main result (see Section 2) tells how the line bundles L intersect
x
the F-curves in M . This allows us to describe a finite set of generators for the
0,n
GIT cone, which we call the GIT complex 0-cells. We show that for n≥6 GC is
strictly smaller than the nef cone.
ThesymmetricgroupS actsonM bypermutingthepoints,andsymmetric
n 0,n
divisors often play a key role. Let S denote the symmetrization of L , that is,
x x
N
S := L . WecalltheconegeneratedbythesethesymmetrizedGITcone,
x σ∈Sn σx
and denote it SGC. We give a conjectural list of extremal rays of SGC and give
an example showing that SGC is strictly smaller than the symmetric nef cone.
Thus, the L and S do not completely describe the Mori theory of M .
x x 0,n
Nevertheless, they contain many geometrically interesting divisors, including the
classes ψ and κ, and we believe these cones are very useful for the study of M .
i 0,n
As an illustration, we reprove a result of Simpson and Fedorchuk–Smyth, also
recently proved by Kiem–Moon.
Matthew Simpson proved the following results in his dissertation: Assume the
S -equivariant Fulton Conjecture (which is known for n ≤ 24 [10]). If certain
n
divisors A (defined below) on M are nef, then the log canonical models of
α 0,n
the pair (M ,∆) can be identified either as M for symmetric sets of weights
0,n 0,β
β or as (P1)n//SL(2) with the symmetric linearization (cf. [24] Corollary 2.3.5,
Theorem2.4.5.) SimpsonthenprovesthatthesedivisorsA areF-nef([24]Propo-
α
sition 2.4.6) and hence, assuming the S -equivariant Fulton Conjecture, nef. As a
n
corollary, the log canonical models are indeed the spaces claimed.
InSection3,weexpressSimpson’sdivisorsA asexpliciteffectivecombinations
α
of a small number of the L . In particular, this proves that Simpson’s A are nef
x α
without using the S -equivariant Fulton Conjecture. In Section 4, we prove that
n
A is ample on the appropriate M . We thus reprove Simpson’s result without
α 0,β
using the S -equivariant Fulton Conjecture.
n
Remark. Fedorchuk and Smyth ([7]) and Kiem and Moon ([21]) have given
proofs of Simpson’s result which are also fully independent of the S -equivariant
n
Fulton Conjecture.
2. The main line bundles and their intersection theory
LetxbeavectorinQnsuchthat0<x ≤1foralliandPn x =2. Weassociate
i i=1 i
a line bundle L to x in the following way: Let ‘ be the least common multiple
x
of the denominators of the x , and write x˜ = ‘x . Since the group SL(2) has no
i i i
nontrivialcharacters,thelinebundleO(x˜ ,...,x˜ )hasauniquelinearization,and
1 n
theGITquotient(P1)n// SL(2)hasadistinguishedamplelinebundleL . Onthe
x x
otherhand,xgivesalinearizationfortheactionofT =Gn/diagG onGr(2,n),
m m
and by [17] one has the Gelfand-MacPherson correspondence (P1)n// SL(2) ∼=
x
Gr(2,n)// T (see [15] for a nice exposition). The dependence of the quotient on
x
Nef divisors on M from GIT 3
0,n
the linearization is called variation of GIT and is described in [3], [26].
There is a birational morphism π :M →(P1)n// SL(2) ([18], [17], [13]).
x 0,n x
Definition 2.1. L :=π∗L .
x x x
(As the referee remarked to us, GIT quotients can also be taken with respect
to semiample line bundles. Thus, this definition makes sense even if some x = 0
i
(with the caveat that π is no longer birational).
x
The line bundle L is nef, since it is the pullback of an ample bundle under a
x
morphism. If 0 < x < 1 for all i = 1,...,n, then the morphism π is birational,
i x
and then L is also big.
x
We can extend the definition to cover the case when some coordinates of x are
zero. Start with n=4. Then if any x =0, L is the trivial line bundle. If n>4
i x
and x = 0, then define L = π∗L , where π : M → M is the morphism
i x i xˆ i 0,n 0,n−1
whichforgetstheith markedpointandxˆ isthevectorobtainedfromxbyomitting
the ith coordinate. We also formally set L =0 if any x =1.
x i
The following lemma allows us to compute intersections of the L with F-
x
curves:
Lemma 2.2. Let {a,b,c,d} be a partition of the set {1,...,n} into four nonempty
subsets,andletF bethecorrespondingF-curveclass. Letxbeasetofweights
a,b,c,d
P P
as above. Write x = x , x = x , etc. We abbreviate
a i∈a i b i∈b i
min:=min{x ,x ,x ,x }, max:=max{x ,x ,x ,x }.
a b c d a b c d
Then

0 if max≥1,

L ·F = 2(1−max) if max≤1 and max+min≥1, (1)
x a,b,c,d
 2min if max≤1 and max+min≤1.
Proof. We reduce the calculation to M and exploit the fact that M ∼= P1
0,4 0,4
is in a natural way a conic in the toric variety P2. For simplicity of notation,
we assume that the partition {a,b,c,d} is ordered, i.e. a = {1,...,#a}, b =
{#a+1,...,#a+#b}, etc.
Recall([19],[20])thatanF-curveinM parametrizesnodalcurveswhosedual
0,n
graph is a trivalent tree except at one vertex, which is 4-valent. As the cross ratio
of the four nodes on the corresponding component varies, a P1 in M is swept
0,n
out. The 4-valent vertex partitions the n leaves into 4 groups, and this is recorded
by a partition {a,b,c,d}. Two F-curves are linearly equivalent if and only if they
have the same partition, so the partition specifies a class F .
a,b,c,d
Now consider the map (P1)4 → (P1)n which is the product of the diagonal
map ∆ : P1 → (P1)a and the similar maps ∆ ,∆ ,∆ . The pullback of the
a b c d
linearizationxon(P1)n to(P1)4 isx0 =(x ,x ,x ,x ). Therefore, thepullbackof
a b c d