Applied Mori theory of the moduli space of stable pointed rational curves [Elektronische Ressource] / Paul L. Larsen. Gutachter: Klaus Altmann ; Gavril Farkas ; Angela Gibney

Paul L. Larsen M.Sc., B.A., B. S.

Sujets

• Mathematics

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 27
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Applied Mori theory of the moduli space of stable
pointed rational curves
DISSERTATION
zur Erlangung des akademischen Grades
Dr. rer. nat.
im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät II
Humboldt-Universität zu Berlin
von
Paul L. Larsen M.Sc., B.A., B.S.
Präsident der Humboldt-Universität zu Berlin:
Prof. Dr. Dr. h.c. Christoph Markschies
Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II:
Prof. Dr. Peter Frensch
Gutachter:
(i) Prof. Dr. Klaus Altmann
(ii) Prof. Dr. Gavril Farkas
(iii)Prof. Dr. Angela Gibney
eingereicht am: 21. Juli 2010
Tag der mündlichen Prüfung: 9. November 2010To my grandparents, Wilbur and Cleta.Acknowledgments
I would like to thank my advisor, Gavril Farkas, first and foremost. His wealth of ideas
and explanations, as well as his support across two continents of thesis work, have been
invaluable. I am indebted to several mathematicians, conversations with whom helped
shape the three main parts of this thesis. Chapter 2 on Fulton’s conjecture benefitted
from conversations with Frank Sottile and Günter Ziegler, the role of toric geometry
in Chapters 3 and 4 originated in conversations with Nathan Ilten, who also answered
innumerable questions about toric geometry, and input from Sam Payne on Chapter 4
led to several improvements.
I am very fortunate to have benefitted from the supportive and productive atmosphere
of the mathematics departments at the University of Texas at Austin, where my Ph.D.
studies began, and at the Humboldt-Universität zu Berlin. I am especially grateful to
Filippo Viviani, who was always willing to hear my questions, and always ready with a
detailed answer. I would like to thank Fabian Müller and Hendrik Süß for proofreading
parts of the thesis. I am grateful to Mark Blume both for pointing out the connection
between Chapter 3 and Losev-Manin moduli spaces and for subsequent discussions.
I have benefitted from conversations with Anil Aryasomayajula, Remke Kloosterman,
Margherita Lelli-Chiesa, Cristina Manolache, and Nicola Tarasca. Dave Jensen and
Kartik Venkatram have also been valuable sounding boards and resources during the
past years. It is a pleasure to thank Dennis Groh of the Computer- und Medienservice
at the Humboldt-Universität, who provided significant help with the figures and other
ALT Xissues.E
I would like to gratefully acknowledge the support of the Harrington and NSF-VIGRE
Fellowships at the University of Texas at Austin. I am very grateful to Dan Freed, who
kindly supported me from his NSF Grant DMS-0603964 in the summer of 2007.
My parents, Karen and Arnold Larsen, have given me more than I could ever acknowl-
edge. Their support and encouragement over the years are the base upon which I have
tried to build. I would like to thank Terezija and Janez Frelih for giving me a home in
Europe, and for providing a work environment to rival Oberwolfach. I am also indebted
to Jim and Sue Calhoun, and Greg Porter for their support and prayers through the ups
and downs of the thesis process.
The youngest person I would like to thank is my daughter, Lucija. In the past year
and a half, Lucija has been never predictible, on occasion my mathematical muse, and
always a joy. My gratitude towards Ana, my wife, has grown month by month and year
by year. She has been the best companion I could possibly hope for during my Ph.D.
studies, sharing with me her wisdom, understanding, and laughter.
vAbstract
We investigate questions motivated by Mori’s program for the moduli space of
stable pointed rational curves, M . In particular, we study the nef cone of M0,n 0,n
(Chapter 2), the Cox ring of M (Chapter 3), and the cone of movable curves of0,n
M (Chapter 4).0,6
In Chapter 2, we prove Fulton’s conjecture forM ,n≤ 7, which states that any0,n
divisor on M non-negatively intersecting all members of a distinguished, finite0,n
collection of curves, called F-curves, is linearly equivalent to an effective integral
sum of boundary divisors. As a corollary, it follows that a divisor on M is nef if0,n
and only if the divisor intersects all F-curves non-negatively. By duality, we thus
recover Keel and McKernan’s result that the F-curves generate the closed cone of
curves ofM forn≤ 7, but with methods that do not rely on negativity properties0,n
of the canonical bundle that fail for higher n.
Chapter 3 initiates a study of relations among generators of the Cox ring ofM .0,n
We first prove a ‘relation-free’ result that exhibits polynomial subrings of Cox(M )0,n
inboundarysectionvariables. IfG isacollectionofboundarydivisorsections, onebnd
for each boundary divisor, and if φ : C[G ]→ Cox(M ) is the homomorphismbnd 0,n
sending a b section to its image in the Cox ring, then these polynomial
subrings in Cox(M ) are the image underφ of subrings having trivial intersection0,n
with the kernel of φ, i.e. they meet the ideal of relations trivially. In the opposite
direction, we exhibit multidegrees in Cox(M ) such that the corresponding graded0,n
parts ofC[G ] meet the ideal of relations non-trivially, hence giving ‘relation-full’bnd
collections in the Cox ring.
In Chapter 4, we study the so-called complete intersection cone of the threefold
M . For a smooth projective varietyX, this cone is defined as the closure of curve0,6
classes obtained as intersections of dim(X)− 1 very ample divisors. The complete
intersection cone is contained in the cone of movable curves, which, by results of
Boucksom, Demailly, Pˇaun, and Peternell, is dual to the cone of pseudoeffective
divisors. We show that, for a series of toric birational models for M related to0,6
the Kapranov blow-up construction, the complete intersection and movable cones
coincide, while for M , there is strict containment of these cones.0,6
Key words: Algebraic geometry, moduli spaces of curves, birational geometry,
Mori theory, toric varieties, combinatorics
viiZusammenfassung
Diese Dissertation befasst sich mit Fragen über den Modulraum M der stabi-0,n
len punktierten rationalen Kurven, die durch das Mori-Programm motiviert sind.
Insbesondere studieren wir den nef-Kegel von M (Chapter 2), den Cox-Ring von0,n
M (Chapter 3), und den Kegel der beweglichen Kurven von M (Chapter 4).0,n 0,6
In Kapitel 2 beweisen wir Fultons Vermutung für M , n≤ 7. Diese Vermutung0,n
besagt, dass ein Divisor, der alle Elemente einer gewissen endlichen Kollektion von
Kurven, sogenannteF-Kurven, mit nichtnegativer Multiplizität schneidet, als effek-
tive ganzzahlige Linearkombination von Randdivisoren dargestellt werden kann. Als
Korollar folgt, dass ein Divisor genau dann nef ist, wenn die entsprechende Schnitt-
multiplizität mit jeder F-Kurve nichtnegativ ist. Mittels Dualität bekommen wir
dadurch einen neuen Beweis des Resultats von Keel und McKernan, dass die F-
Kurven für n≤ 7 den Kegel der Kurven von M erzeugen, jedoch mit Methoden,0,n
die unabhängig von Negavititätseigenschaften des kanonischen-Bündels sind, welche
für größeres n nicht mehr stimmen.
Kapitel 3 beginnt ein Studium der Relationen zwischen erzeugenden Elemen-
ten des Cox-Rings von M . Wir beweisen zuerst einen “Relationenfreiheitssatz”,0,n
der in Cox(M ) polynomiale Unterringe indentifiziert. Ist G eine Kollektion0,n bnd
von Schnitten von Randdivisoren, einer für jeden Randdivisor, und bezeichnet φ :
C[G ]→ Cox(M ) den Homomorphismus, der ein zu einem Randdivisor gehö-bnd 0,n
riges Element ausC[G ] auf sein Bild in Cox(M ) schickt, dann ist jeder dieserbnd 0,n
polynomialenUnterringein Cox(M )dasBildunterφvoneinemUnterring,dessen0,n
Schnit mit dem Kern von φ trivial ist, das heißt, der Unterring schneidet das Ideal
der Relationen trivial. In der anderen Richtung geben wir Multigrade in Cox(M )0,n
an, sodass der zuhegörige graduierte Tel vonC[G ] das Ideal der Relationen nicht-bnd
trivial schneidet und erhalten so in Resultat über “volle Relationen”.
In Kapitel 4 studieren wir den sogennanten Kegel der vollständigen Durchschnit-
te der 3-Fläche M . Für eine glatte projektive Varietät X ist dieser Kegel als0,6
Abschluss der Kurven definiert, die sich als Durchshnitt von dim(X)− 1 sehr am-
plen Divisoren darstellen lassen. Der Kegel der vollständigen Durchschnitte ist ein
Unterkegel des Kegels der beweglichen Kurven. Nach Ergebnissen von Boucksom,
Demailly,Pˇaun,undPeternellistderbeweglicheKegeldualzumKegelderpseudoef-
fektiven Divisoren. Wir beweisen für eine Reihe von torischen birationalen Modellen
vonM , dass der Kegel der vollständigen Durchschnitte und der bewegliche Kegel0,6
übereinstimmen, während für M die Inklusion echt ist.0,6
Schlagworte: Algebraische Geometrie, Modelräume von Kurven, Birationelle Geo-
metrie, Mori-Theorie, Torische Varietäten, Kombinatorik
ix