Table Of ContentClay Mathematics Proceedings
Volume 18
A Celebration of Algebraic
Geometry
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Brendan Hassett
James McKernan
Jason Starr
Ravi Vakil
Editors
American Mathematical Society
Clay Mathematics Institute
A Celebration of Algebraic
Geometry
Clay Mathematics Proceedings
Volume 18
A Celebration of Algebraic
Geometry
A Celebration of Algebraic Geometry:
A Conference in Honor of Joe Harris’ 60th Birthday
Harvard University
Cambridge, MA
August 25–28, 2011
Brendan Hassett
James McKernan
Jason Starr
Ravi Vakil
Editors
American Mathematical Society
Clay Mathematics Institute
2010 Mathematics Subject Classification. Primary 14C30, 14D22, 14H45, 14J28, 14K10,
14N15, 14N20.
Cover photo of Joe Harris is by David Mosher, copyright the Simons Foundation.
Library of Congress Cataloging-in-Publication Data
Acelebrationofalgebraicgeometry: inhonorofJoeHarris’60thbirthday,August25–28,2011,
HarvardUniversity,Cambridge,Massachusetts/BrendanHassett,JamesMcKernan,JasonStarr,
RaviVakil,editors.
pagescm. —(Claymathematicsproceedings: volume18)
“ClayMathematicsInstitute.”
Includesbibliographicalreferences.
ISBN978-0-8218-8983-1(alk.paper)
1.Geometry,Algebraic. I.Harris,Joe,1951– II.Hassett,Brendan. III.McKernan,James,
1964– IV. Starr, Jason M. (Jason Michael) V. Vakil, Ravi. VI. American Mathematical
Society. VII.ClayMathematicsInstitute.
QA564.C425 2013
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2013014118
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To Joe Harris, for all he has taught us over the years
Contents
Introduction ix
Classification of 2-Fano manifolds with high index
Carolina Araujo and Ana-Maria Castravet 1
Abelian varieties associated to Gaussian lattices
Arnaud Beauville 37
Rank of divisors on graphs: An algebro-geometric analysis
Lucia Caporaso 45
A tour of stable reduction with applications
Sebastian Casalaina-Martin 65
The solvable monodromy extension property and varieties of log general type
Sabin Cautis 119
Effective divisors of moduli spaces of curves and abelian varieties
Dawei Chen, Gavril Farkas, and Ian Morrison 131
Rational self maps of Calabi-Yau manifolds
Xi Chen 171
Variations on Nagata’s conjecture
Ciro Ciliberto, Brian Harbourne, Rick Miranda, and Joaquim
Ro´e 185
Symplectic restriction varieties and geometric branching rules
Izzet Coskun 205
Riemann-Roch for Deligne-Mumford stacks
Dan Edidin 241
The regularity of the conductor
David Eisenbud and Bernd Ulrich 267
Stability of genus five canonical curves
Maksym Fedorchuk and David Ishii Smyth 281
Restriction of sections for families of abelian varieties
Tom Graber and Jason Michael Starr 311
Correspondence and cycles spaces: A result comparing their cohomologies
Mark Green and Phillip Griffiths 329
vii
viii CONTENTS
Geometry of theta divisors—a survey
Samuel Grushevsky and Klaus Hulek 361
Singular curves and their compactified Jacobians
Jesse Leo Kass 391
On the G¨ottsche threshold
Steven L. Kleiman and Vivek V. Shende,
with an appendix by Ilya Tyomkin 429
Curve counting a` la Go¨ttsche
Steven L. Kleiman 451
Mn¨ev-Sturmfels universality for schemes
Seok Hyeong Lee and Ravi Vakil 457
Gromov-Witten theory and Noether-Lefschetz theory
Davesh Maulik and Rahul Pandharipande 469
Numerical Macaulification
Juan Migliore and Uwe Nagel 509
The non-nef locus in positive characteristic
Mircea Mustat¸a˘ 535
Pairwise incident planes and hyperka¨hler four-folds
Kieran G. O’Grady 553
Derived equivalence and non-vanishing loci
Mihnea Popa 567
Degenerations of rationally connected varieties and PAC fields
Jason Starr 577
Remarks on curve classes on rationally connected varieties
Claire Voisin 591
Introduction
Joseph Daniel Harris, universally known as “Joe” in the algebraic geometry
community, celebrated his 60th birthday in 2011. On 25–28 August 2011, an al-
gebraic geometry conference, “A Celebration of Algebraic Geometry”, was held at
Harvard University with over 300 mathematicians participating.
This milestone was cause for celebration not only for Joe, but for all of us
who have been fortunate to know and work with Joe over the years, and for the
many more who have been touched by Joe’s research, his books, and his vivacious
lectures.
This is not the moment to catalogue Joe’s numerous accomplishments and ac-
colades. However,wewouldliketorecallsomeofJoe’smostlastingimpactsonour
subject. Joe Harris was both an undergraduate and graduate student at Harvard
University, completing his Ph.D thesis with Phillip Griffiths in 1978. During those
student years, Griffiths and Harris wrote together their textbook in complex ge-
ometry, “Principles of Algebraic Geometry”. This has been the textbook for both
complex algebraic geometry and complex differential geometry ever since, as well
as the source of many “Joe legends”.
Joehasover100publishedresearcharticles. Fromthebeginning,Joewasasu-
perstar in the subject of algebraic curves, particularly the moduli spaces of curves.
Signal achievements were Joe’s completion with Griffiths of the Brill-Noether the-
orem; the proof with Mumford, and then Eisenbud, that moduli spaces of curves
areof general typeforg >23; themany consequencesofthe theoryof “limit linear
series” of Eisenbud-Harris; and the proof of the Severi conjecture on irreducibility
of Severi varieties of plane curves. But this captures only a small part of Joe’s
work: he has also worked on local differential geometry of algebraic varieties, vari-
ations of Hodge structure, extension of Thom-Porteous theory to the symmetric
andskew-symmetricsetting, varietiesofminimal degree, Noether-Lefschetztheory,
the Harris-Morrison slope conjecture, Cayley-Bacharach theory, the uniformity of
rationalpointsofcurvesovernumberfields, theenumerativetheoryofplanecurves
with specified contact orders (i.e., relative Gromov-Witten invariants of the pro-
jectiveplane, in modernparlance), unirationality oflow degree varieties, rationally
connectedvarieties,andmodulispacesofrationalcurvesonFanomanifolds,among
many others.
Formanyofus,equallyremarkableisJoe’sunflaggingsupportofyoungermath-
ematicians, reflected in his teaching and his mentoring. Joe served as the graduate
program director of the Harvard mathematics department for many years, shep-
herding generations of graduate students. Joe was the mentor for many Benjamin
Peirce instructors and other postdoctoral fellows. Joe has been the thesis advisor
ix
