Algebraic Geometry Seminar

Department of Mathematics
Princeton University

Spring 2013 Lectures

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 3:30)
Place: Fine 322

 Date Speaker Title Feb 5 Zachary Maddock Columbia University Regular del Pezzo surfaces with irregularity Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. Â We construct the first examples of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_XÂ ) > 0. Â Our construction is by quotienting a regular, quasi-linear surface (i.e. a regular variety that is geometrically a non-reduced first-order neighborhood of a plane) by explicit rank 1 foliations. We also find a restriction on the integer pairs that are possible as the anti-canonical degreeÂ and irregularityÂ of such surfaces. Feb 12 Joseph Ayoub Zurich University Motivic Galois groups and periods We explain how to associate a universal pro-algebraic group to the Betti realization functor from the triangulated category of motives over a subfield of $\mathbb{C}$. We then give a concrete description of the torsor of isomorphisms between the Betti realization and the de Rham realization. If time permits, some applications to periods will be sketched. Feb 19 John Lesieutre Massachusetts Institute of Technology A divisor with non-closed diminished base locus I will explain the construction of a pseudoeffective R-divisor Dλ on the blow-up of P3 at nine very general points which has negative intersections with an infinite set of curves, whose union is Zariski dense. Â It follows that the diminished base locus B-(Dλ) = ∪A ample B(Dλ+A) is not closed and that Dλ does not admit a Zariski decomposition in even a very weak sense. Â Along the way I will discuss some related examples, including an R-divisor which is nef on very general fibers of a family, but fails to be nef over countably many prime divisors in the base. Feb 25, 2:30-3:30 Yuji Odaka Kyoto University Algebraic Geometry of K-stability and its application to Moduli varieties K-stability is a stability for varieties (Tian, Donaldson), a modification of classical stability (Mumford). While it has been known for several decades that classically stability does not work in higher dimensions moduli construction, the author explains how the K-stability fits into recent construction of compact moduli of general type varieties by KSBA (Koll\'ar-Shepherd-Barron, Alexeev) theory. As the KSBA theory depends on MMP-based birational geometry, not on GIT, it also reflects more general compatibility between the K-stability and such birational algebraic geometry which really exists. If time permits, I explain some partial progress towards my dream "K-moduli", more general moduli via K-stability and cscK metrics. Feb 26 Brian Lehmann Rice University Big cycles and volume functions The volume of a divisor is an important invariant measuring the "positivity" of its numerical class. I will discuss an analogous construction for cycles of arbitrary codimension. In particular, this yields geometric characterizations of big cycle classes modeled on the well-known criteria for divisor and curve classes. Mar 5 Matt Baker Georgia Institute of Technology Which morphisms between tropical curves come from algebraic geometry? A finite morphism between stable marked curves over a non-archimedean field gives rise in a natural way to a morphism of metric graphs (also known as "abstract tropical curves"). Â We study the question of which morphisms between abstract tropical curves arise from this construction. Â This leads us naturally to the notion ofÂ metrized complexes of curves (which are tropical curves endowed with some extra structure) and of harmonic morphisms between metrized complexes. Â We show using Berkovich's theory of analytic spaces that every tamely ramified finite harmonic morphism of metrized complexes of curves arises from aÂ morphism of algebraic curves. Â This generalizes and provides new analytic proofs of earlier results of Saidi and Wewers. Â We also present counterexamples to some possible strengthenings of this result; for example, the gonality of a tropical curve can be strictly smaller than the gonality of any smooth proper lift. Â As an application of the above considerations, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Neron models, providing a negative answer to a question of Ribet. Mar 12 Han-Bom Moon University of Georgia Birational geometry of $\bar{M}_{0,n}$ and conformal blocks In the last several decades, conformal blocks have been studied by many algebraic geometers who are interested in the geometry of moduli spaces of vector bundles. Recently, people have begun to study the relation between conformal blocks and the birational geometry of the moduli space $\bar{M}_{0,n}$ of stable pointed rational curves, because they give a huge family of base point free divisors on $\bar{M}_{0,n}$. In this talk, I will discuss an example of interaction between three approaches to the birational geometry of $\bar{M}_{0,n}$: stack theoretic viewpoint, GIT, and conformal blocks. This is based on a joint work with A. Gibney, D. Jensen and D. Swinarski. Mar 19 (Spring recess) Mar 27 WEDNESDAY, 3:30 NOTE: SPECIAL DAY AND TIME Yukinobu Toda Kavli Institute for the Physics and Mathematics of the Universe Gepner type stability conditions on graded matrix factorizations I will introduce Gepner type Bridgeland stability conditions on graded matrix factorizations. In the case of a quintic 3-fold, such a stability condition may correspond to the Gepner point on the stringy Kahler moduli space via Orlov equivalence. Also such a stability condition is important in finding non-trivial relations among DT invariants. I will show the existence of Gepner type stability conditions on graded matrix factorizations in some low degree cases. I also show that a conjectural construction of a Gepner point for a quintic 3-fold leads to a conjectural stronger version of BG inequality for stable sheaves. Apr 2 Alex Küronya Technical University Budapest Title: Finite generation and geography of models There are two main series of examples where some version of Mori's program can be (at least conjecturally) performed: one is the classical minimal model program associated to adjoint divisors, the other is the case of Mori Dream spaces, that is, varieties with finitely generated 'global' coordinate rings. In an attempt to provide a unifying framework, we are led to study certain locally polyhedral decompositions of regions of the space of divisors, often referred to as 'geography of models'. These decompositions correspond to birational models of the underlying variety, and it is an important question how well-behaved these models are. We show that even under strong  finite generation assumptions, a very important property, the Q-factoriality of these models, is not guaranteed in general, and point out the obstacle. It turns out that this obstacle is hidden behind the definitions in the classical case, but the moment we intend to run MMP in a more general setting, it plays a crucial role. This is an account of joint work with Anne-Sophie Kaloghiros and Vladimir Lazic. Apr 9 Letao Zhang Rice University Representation Theory and Hilbert Schemes of Points on K3 Surfaces Let X be a Kaehler deformation of Hilbert scheme of points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X. Also, we derive a generating series for deducing the number of canonical Hodge classes of degree 2n. Apr 16 Zhiyuan Li Standford University Picard groups on moduli space of K3 surfaces The Noether-Lefschetz (NL) divisors on moduli space of quasi-polarized K3 surfaces are the loci where the Picard number is greater than one. Maulik and Pandharipande have conjectured that NL-divisros will span the Picard group of the moduli space. I will talk about this problem from both geometry and arithmetic. In particular, we verify this conjecture via GIT when the degree of the K3 surface is small. I will also talk about the general case and the relation to automorphic representation theory. A conjectural approach to this problem may be discussed at the end of this talk. This is joint work with Zhiyu Tian. Apr 23 TBA TBA TBA TBA Apr 30 Xin Zhou University of Michigan Effective non-vanishing of asymptotic syzygies In this talk, I will first introduce classical results on the asymptotics of syzygies. Then, I will motivate the effective result with recent work of Ein and Lazarsfeld. At the end, I will discuss the implications of the proof of the effective statement for toric varieties. May 7 Zhiyu Tian California Institute of Technology Weak approximation for cubic hypersurfaces Given an algebraic variety X over a field F (e.g. number fields, function fields), a natural question is whether the set of rational points X(F) is non-empty. And if it is non-empty, how many rational points are there? In particular, are they Zariski dense? Do they satisfy weak approximation? For cubic hypersurfaces defined over the function field of a complex curve, we know the existence of rational points by Tsen' s theorem or the Graber-Harris-Starr theorem. In this talk, I will discuss the weak approximation property of such hypersurfaces. May 14 Anand Deopurkar Columbia University Compactifying spaces of branched coversModuli spaces of geometrically interesting objects are often non-compact. They need to be compactified by adding some degenerate objects. In many cases, this can be done in several ways, leading to a menagerie of birational models, which are related to each other in interesting ways. In this talk, I will explore this idea for the spaces of branched covers of curves, known as the Hurwitz spaces. I will construct a number of compactifications of these spaces by allowing more and more branch points to coincide. I will describe the geometry of the resulting spaces for the case of triple covers.

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