Table Of ContentMEMOIRS
of the
American Mathematical Society
Number 985
The Moduli Space of Cubic Threefolds
as a Ball Quotient
Daniel Allcock
James A. Carlson
Domingo Toledo
January 2011 • Volume 209 • Number 985 (fourth of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 985
The Moduli Space of Cubic Threefolds
as a Ball Quotient
Daniel Allcock
James A. Carlson
Domingo Toledo
January2011 • Volume209 • Number985(fourthof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Allcock,Daniel,1969-
The moduli space of cubic threefolds as a ball quotient / Daniel Allcock, James A. Carlson,
DomingoToledo.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 985)
“January2011,Volume209,number985(fourthof5numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4751-0(alk. paper)
1.Modulitheory. 2.Surfaces,Cubic. 3.Threefolds(Algebraicgeometry). I.Carlson,James
A.,1946- II.Toledo,Domingo. III.Title.
QA564.A48 2011
516.3(cid:2)5—dc22 2010037801
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Dedicated to Herb Clemens and Phillip Griffiths
Contents
Introduction ix
Chapter 1. Moduli of Smooth Cubic Threefolds 1
Chapter 2. The Discriminant near a Chordal Cubic 13
Chapter 3. Extension of the Period Map 27
Chapter 4. Degeneration to a Chordal Cubic 39
4.1. Statement of results 39
4.2. Overview of the calculations 42
4.3. Semistable reduction 42
4.4. Cohomology computations 49
Chapter 5. Degeneration to a Nodal Cubic 55
Chapter 6. The Main Theorem 61
Chapter 7. The Monodromy Group and Hyperplane Arrangement 65
Bibliography 67
Index 69
v
Abstract
The moduli space of cubic threefolds in CP4, with some minor birational
modifications, is the Baily-Borel compactification of the quotient of the complex
10-ball by a discrete group. We describe both the birational modifications and the
discrete group explicitly.
ReceivedbytheeditorNovember6,2006.
ArticleelectronicallypublishedonJuly15,2010;S0065-9266(10)00591-0.
2000 MathematicsSubjectClassification. Primary32G20;Secondary14J30.
Key wordsand phrases. Ballquotient,periodmap,modulispace,cubicthreefold.
D. Allcock was partly supported by NSF grants DMS 0070930, DMS-0231585 and DMS-
0600112.
J.A.CarlsonwaspartlysupportedbyNSFgrantsDMS9900543,DMS-0200877andDMS-
0600816andtheClayMathematicsInstitute.
D. Toledo was partly supported by NSF grants DMS 9900543, DMS-0200877 and DMS-
0600816.
(cid:2)c2010 American Mathematical Society
vii
Introduction
One of the most basic facts in algebraic geometry is that the moduli space of
elliptic curves, which can be realized as plane cubic curves, is isomorphic to the
upperhalfplanemodulotheactionoflinearfractionaltransformationswithinteger
coefficients. In [3], we showed that there is an analogous result for cubic surfaces;
the analogy is clearest when we view the upper half plane as complex hyperbolic
1-space,thatis,astheunitdisk. Theresultisthatthemodulispaceofstablecubic
surfacesisisomorphictoaquotientofcomplexhyperbolic4-spacebytheactionofa
specific discrete group. This is the group of matrices which preserve the Hermitian
form diag[−1,1,1,1,1] and which have entries in the ring of Eisenstein integers:
the ring obtained by adjoining a primitive cube root of unity to the integers. The
idea of the proof is not to use the Hodge structure of the cubic surface, which
has no moduli, but rather that of the cubic threefold obtained as a triple cover
of CP3 branched along the cubic surface. The resulting Hodge structures have a
symmetry of order three, and the moduli space of such structures is isomorphic to
complex hyperbolic 4-space CH4. This is the starting point of the proof, which
relies crucially on the Clemens-Griffiths Torelli Theorem for cubic threefolds [10].
The purpose of this article is to extend the analogy to cubic threefolds. The
ideaistousetheperiodmapforthecubicfourfoldsobtainedastriplecoversofCP4
branched along the threefolds, using Voisin’s Torelli theorem [41] in place of that
of Clemens and Griffiths. In this case, however, a new phenomenon occurs. There
is one distinguished point in the moduli space of cubic threefolds which is a point
of indeterminacy for the period map. This point is the one represented by what
we call a chordal cubic, meaning the secant variety of a rational normal quartic
curve in CP4. The reason for the indeterminacy is that the limit Hodge structure
depends on the direction of approach to the chordal cubic locus. In fact, the limit
dependsonly onthisdirection,andsotheperiodmapextendstotheblowupofthe
moduli space.
The natural period map for smooth cubic threefolds [10] embeds the moduli
space in a period domain for Hodge structures of weight three, namely, a quotient
oftheSiegelupperhalfspaceofgenusfive. Forthisembedding,however,thetarget
space has dimension greater than that of the source. For the construction of this
article, the dimensions of the source and target are the same.
Toformulate the mainresult, letM bethe GIT moduli space ofcubic three-
ss
folds, and let M(cid:2) be its blowup at the point corresponding to the chordal cubics.
ss
LetM ⊆M bethe moduli space of stable cubic threefolds, andlet M(cid:2) be M(cid:2)
s ss s ss
minus the proper transform of M −M . Let M be the moduli space of smooth
ss s 0
cubic threefolds. Then we have the following, contained in the statement of the
main result, Theorem 6.1:
ix
