## Ohio State University Algebraic Geometry Seminar## Year 2013-2014Time: Tuesdays 3-4pmLocation: MW 154 |
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(Bayer): I will report on joint work with T. Bridgeland, in which we give a complete description of the group of autoequivalences of the derived category of K3 surfaces with Picard rank one. This result is based on a geometric argument: we show that the space of stability conditions is simply-connected.

(Fusi): In a joint work with Tommaso de Fernex, we prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is a countable union of closed subsets of the parameter space.

(Schmidt): Bridgeland stability conditions are an adaption of the concept of slope stability to triangulated categories. I will discuss a successful method for their construction on the bounded derived category of the smooth quadric threefold. This is closely related to a generalized Bogomolov-Gieseker inequality involving third Chern characters that was conjectured for any smooth projective threefold by Bayer, Macrì, and Toda.

(Degeratu): A Calabi-Yau orbifold is locally modelled on C^n/G with G a finite subgroup of SU(n). A crepant resolution of such a singularity is a resolution with trivial canonical bundle. In this talk I will show how an interplay between analytic and algebraic techniques can be used to obtain insights about the geometry and topology of crepant resolutions in complex dimension 3. Specifically, we prove a higher dimensional analogue of Kronheimer and Nakajima's geometrical McKay correspondence. This is joint work with Thomas Walpuski.

(Tevelev): I'll report on a joint project with Hacking and Urzua, where we continue the study of flips of semistable extremal neighborhoods started by Mori, Kollár, and Prokhorov. Our motivation is to make explicit the computation of stable one-parameter degenerations of smooth canonically polarized surfaces introduced by Kollár and Shepherd-Barron.

(Sheshmani): We construct an algebraic-geometric framework to calculate the partition functions of "massive black holes" enumerating invariants of supersymmetric D4-D2-D0 BPS states in type IIA string theory. Using S-duality, the entropy of such black holes can be related to a certain N=2, d=4 Super Yang-Mills theory on a divisor in a threefold. Physicsts: Gaiotto, Strominger, Yin, Denef, Moore, via careful study of such S-duality, have conjectured that these partition functions have modular properties. We give a rigorous mathematical proof of their conjectures in different geometric setups. This is a report of joint project with Amin Gholampour and Richard P. Thomas. We also use an algebro-geometric analogue of the string theoretic D4/D2 T-duality to prove the modularity properties of certain PT stable pair invariants over threefolds given by smooth and Nodal surface fibrations over a curve. Here our strategy is to use combination of degeneration techniques, conifold transitions, and wall crossing of Bridgeland stability conditions. This is a report of joint project with Gholampour and Yukinobu Toda.

(Gallardo): We describe the use of GIT and stable replacement for studying the geometry of a special compactification of the moduli space of smooth quintic surfaces, the KSBA compactification. In particular we discuss the interplay between non-log-canonical singularities and boundary divisors. Our presentation will be enriched by drawing analogies with similar phenomena on the moduli space of curves of genus three.

(Heidersdorf): I will talk about the structure of the tensor category Rep(Gl(m|n)) where Gl(m|n) is the General Linear Supergroup on m even and n odd variables. The category Rep(Gl(m|n) is not semisimple and the decomposition of tensor products into the indecomposable constituents is only known in special cases. Every reasonably nice tensor category C has a unique proper tensor ideal N such that the quotient C/N is a semisimple tensor category. I will apply this construction to the tensor category C = Rep(Gl(m|n)). This quotient is again the representation category of a pro-reductive Supergroup scheme. I will show some results about this Supergroup scheme and what this implies about tensor product decompositions.

(Markman): Let X be a smooth complex projective variety and c a monodromy invariant cohomology class on X. The Hodge conjecture predicts that c is algebraic. One way to prove it would be to exhibit c as a characteristic class of a coherent sheaf on X. When X is hyperkahler there is a very powerful deformation theoretic technique, due to Verbitsky, that realizes c as a characteristic class for all deformations of X, once it is realized as a characteristic class of a stable sheaf with monodromy invariant Chern classes on one Y deformation equivalent to X. We will discuss an example, which led to the proof of the standard conjectures for hyperkahler varieties of K3^[n]-type (joint with F. Charles) and which is central in the study of generalized (non-commutative and gerby) deformations of K3 surfaces (joint with S. Mehrotra).

(Basu): We shall discuss the enumerative problem of counting the number of complex curves (in complex projective space of dimension 2) which pass through the requisite number of generic points and has a prescribed singularity at one point. Our exposition will be from a topological point of view via the ubiquitous Euler class.

(Tirabassi): We give a cohomologoical characterization of products of theta divisors among irregular varieties with Euler characteristic one. We also provide a complete classification of varieties X with chi equal one, irregularity 2 dim X -1, extending some results of Hacon--Pardini. This is joint work with Z. Jiang and M. Lahoz.

(Tseng): The goal of this (largely expository) talk is to explain: i) the idea that Gromov-Witten invariants (which count curves) are expected to obey integrable systems of partial differential equations; ii) an approach to construct explicit integrable systems governing Gromov-Witten invariants of a given target space, via mirror symmetry; iii) Virasoro constraints.

(Sheshmani): I will talk about joint work with Toda and Gholampour on studying the stable pair theory of K3 fibrations over curves with possibly nodal fibers. We express the stable pair invariants of the fiberwise irreducible classes in terms of the famous Kawai-Yoshioka formula for the Euler characteristics of moduli space of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants to the perverse (non-commutative) stable pair invariants of the K3 fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants of the fiberwise curve classes in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers. Finally if time permits, I will discuss briefly the work in progress, where we use the non-commutative stable pairs theory to prove the famous KKV conjecture.

(Fulghesu): There is a well developed intersection theory on quotient stacks obtained by algebraic schemes acted on by algebraic groups. Moduli stacks have usually a presentation as quotient stacks. Unfortunately, determining the integral intersection ring is, in general, quite difficult. However, if we can represent a stack as a quotient by GLn (or PGL2), the task becomes easier. This is, for example, the case of the Artin stack of reduced quadrics and the stack of cyclic covers of the projective line.

(Shipman): The bounded derived category D(X) of a variety X encodes all homological properties of sheaves on the variety. At first glance, D(X) seems large and unwieldy, but in fact it can be computed in a sense and behaves much like the cohomology of a variety. There are many examples where D(X) is equivalent to the derived category of X' where X' is another variety, a stack, or a module category for a (noncommutative) algebra. Given an equivalence D(X) ~ D(X') we can view the category coh(X') as a subcategory of D(X), leading to the question: what is the relationship between coh(X') and coh(X)? I will explain recent work with Morgan Brown on how in dimension 2 one can exploit stability and Harder-Narasimhan filtrations on X' to construct coh(X') (as a subcategory of D(X)) from coh(X).

(Brown): Let G be a subgroup of SL_n(\C). When n=2, \C^n/G has a unique minimal resolution, and the classical McKay correspondence relates the representation theory of G with the structure of this resolution. For n=3, Bridgeland, King, and Reid used categorical techniques to show that \C^n/G$ has a distinguished crepant resolution Y=G-Hilb. Specifically, they showed that there is an equivalence between the derived categories D^G(\C^n) and D(Y). I will show how one can use a notion of stability to describe coherent sheaves on Y in terms of complexes of G-equivariant objects on \C^n.

(Silberstein): We present the technique of geometric reconstruction of function fields, and applications to the program of anabelian geometry over algebraically closed fields begun by Bogomolov.

(Gibney): I'll speak about recent joint work with Prakash Belkale and Swarnava Mukhopadhyay where we study divisor classes on the moduli space of curves using tools from quantum cohomology. I will describe some of our results in the context of guiding conjectures about the moduli space of curves. I'll focus on the basic open problem of determining necessary and sufficient conditions for these classes to be nonzero, including an answer in the first nontrivial case.

(Centeleghe): Thanks to an old result of Deligne, the category of ordinary abelian varieties over a fixed finite field can be described in terms of finite free Z-modules equipped with a linear operator F (playing the role of Frobenius) satisfying certain axioms. In a recent joint work with Jakob Stix, we prove a similar result for the full subcategory of all abelian varieties over the prime field F_p given by those objects not admitting a non trivial morphism to a certain simple surface. In the talk I will describe the method we use, which is completely different from that of Deligne.

(Gorsky): A conjecture of Oblomkov and Shende, recently proven by Maulik, relates the Hilbert schemes of points on a plane curve singularity to the invariants of the corresponding knots. I will describe the homology of these Hilbert schemes for singularities {x^n=y^m} corresponding to torus knots, and relate it to the bigraded deformation of Catalan numbers introduced by Garsia and Haiman. The talk is based on joint works with M. Mazin and A. Negut.

(Chen): The theory of stable log maps was developed recently for studying the degeneration of Gromov-Witten invariants. In this talk, I will introduce another important aspect of stable log maps as a useful tool for investigating A^1-curves on quasi-projective varieties, which are the analogue of rational curves on proper varieties. At least two interesting applications of A^1-curves will be introduced in this talk. For classical birational geometry, the A^1-curves can be used to produce very free rational curves on Fano complete intersections in projective spaces. On the arithmetic side, A^1-connectedness gives a general frame work for the existence of integral points over function field of curves. This is joint work with Yi Zhu.

(Akhtar): The edge ideal of a graph, first defined by R. Villareal, is an ideal in a certain polynomial ring over a field, which encodes the combinatorial structure of the graph. Being a finitely generated module over a graded ring, it is natural to study the (graded) Betti numbers of this ideal, and this topic has received considerable attention in the literature. Recently, there has been some interest on the part of algebraic geometers in computing Betti numbers of the higher secant powers of the edge ideal; these are the ideals which define the secant varieties corresponding to the variety defined by the edge ideal. This talk will focus on a recursive formula which gives the Betti numbers of (the secant powers of the edge ideal of) the join of a graph with an independent set, in terms of the Betti numbers of the original graph. This formula will be applied to compute all the Betti numbers in the case of complete graphs and complete multipartite graphs, recovering earlier results of Jacques.

(Pappas): We will explain how to formulate and prove a Riemann-Roch type theorem for bundles on families of curves that uses the theory of higher dimensional adeles of Parshin and Beilinson.

(Stellari): It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. This is a joint work in progress with A. Canonaco.

(Jensen): Many problems in algebraic geometry concern the behavior of linear series on general curves. The recently developed theory of specialization of linear series from algebraic to tropical curves provides us with a new set of tools for approaching such problems. We will show how such tools can be used to produce a new proof of the Gieseker-Petri Theorem, which says that the varieties parameterizing linear series on a general curve are smooth.

(Giansiracusa): The moduli space \bar{M}_{0,n}, a compactification of the space of n distinct points on the Riemann sphere, has served as a fertile testing ground to explore many phenomena of moduli spaces in algebraic geometry. One tantalizing question is to describe the convex cone of effective divisor classes and Cox ring of these spaces. I'll discuss joint work with B. Doran and D. Jensen in which we provide a new perspective on this question in terms of simplicial complexes and show how this relates to recent exciting work of Castravet, Tevelev, and Opie.

This page is maintained by Davide Fusi and Emanuele Macrì.