Algebraic Geometry Seminar

We note that under certain conditions, the area bounded by the logarithmic image of a real plane curve is a half-integer multiple of pi square. The half-integer number can be interpreted as the quantum index of the real curve and used to refine enumerative invariants.

Let F_k(M) denote the ordered k-point configuration space of a connected open manifold M. Work of Church and others shows that for a given manifold, as k increases, this family of spaces exhibits a phenomenon called homological "representation stability" with respect to the natural symmetric group actions. In this talk I will explain what this means, and describe a higher-order "secondary stability" phenomenon among the unstable homology classes. The project is work in progress, joint with Jeremy Miller.

The goal of this talk is to study the positivity of subvarieties with nef normal bundle in terms of intersection theory. We first give a generalization to the notion of subvarieties with nef normal bundle, via the positivity of the exceptional divisor. We call these subvarieties nef. Then we show that restriction of a pseudoeffective divisor to a nef subvariety is pseudoeffective. We also show that nefness is a transitive property. Next, we define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps, and show that this cone contains the movable cone and shares similar intersection-theoretic properties with it, using the aforementioned properties of nef subvarieties. The main tool used in this work is the theory of $q$-ample divisors, as developed by Totaro.

Given a simple Lie algebra $\mathfrak{g}$, a positive integer $\ell$, and an $n$-tuple $\vec{\lambda}$ of dominant integral weights for $\mathfrak{g}$ at level $\ell$, one can define a vector bundle on $\overline{\operatorname{M}}_{g,n}$ known as a \textit{vector bundle of conformal blocks}. These bundles are nef in genus $g=0$ and so this family provides potentially an infinite number of elements in the nef cone of $\M_{0,n}$ to analyze. Result relating these divisors with different data is thus significant in understanding these objects. In this talk, we use correspondences of these bundles with products in quantum cohomology in order to classify when a bundle with $\sL_2$ or $\sP_{2\ell}$ is rank one. We show this is also a necessary and sufficient condition for when these divisors are equivalent.

Let $L$ be a line bundle on a smooth projective surface $X$. A conjecture attributed to S. Mukai says if $L\simeq \omega_{X}\otimes A^{\otimes k}$ for some ample line bundle $A$ and an integer $k\ge 4$, then $L$ is normally generated. In this talk, I will discuss various methods and results in the curve and surface case. I will show the conjecture holds for a double covering over an anticanonical rational surface, which may be viewed as a two dimensional analogue of hyper-elliptic curve, and equivariant ample line bundle $A$. The work builds on an understanding of linear systems on anticanonical rational surfaces, which is largely due to B. Harbourne.

A hyperplane in the vector space k[x_0,...,x_n]_d of homogeneous polynomials of degree d defines the socle of a Gorenstein graded ring. A canonical Bridgeland stability condition allows us to filter these rings (or more precisely their graded resolutions) as objects of the derived category of coherent sheaves and we use this to find some new invariants. In particular, this stratifies the space of polynomials by generalized secant varieties to the Veronese. This is joint work with Brooke Ullery.

Motivated by the study of Kahler-Einstein/Sasaki-Einstein metrics, I will discuss an algebro-geometric problem of minimizing normalized volumes among all real valuations centered at any Q-Gorenstein klt singularity. It was conjectured that there is always a unique minimizer, which should be quasi-monomial. I will discuss recent progresses on this problem, and explain how it is related to the so-called K-semistability and the deFernex-Ein-Mustata type inequalities. Part of this work is based on joint works with Yuchen Liu and Chenyang Xu.

Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model, our work in progress to use it to produce probabilistic information, and our goal of eventually de-tropicalizing our approach and using it to analyze the so-called directed polymer problem.

Work by McQuillan and Brunella demonstrates the existence of a Mori theory for rank 1 foliations on surfaces. In this talk we will discuss an extension of some of these results to the case of rank 2 foliations on threefolds, as well as indicating how a complete Mori theory could be developed in this case.

Following ideas of Marian and Oprea, finite quot schemes can be used to investigate Le Potier's strange duality conjecture for surfaces. I will discuss recent work with Aaron Bertram and Drew Johnson in which we prove the existence of a large class of finite quot schemes on the projective plane. We use nice resolutions of general stable vector bundles, which also yield an easy proof that these bundles are globally generated whenever their Euler characteristic suggests that they should be.

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

There are not many known examples of compact irreducible hyperk?hler manifolds. Two series of examples appear in dimension 2n, for every n>1, and are related to the Hilbert scheme of points on a K3 or an abelian surface; and in dimension 6 and 10 there is one extra, or exceptional, deformation class, each of which was found by O'Grady. While considerable work has been devoted to studying hyperk?hler manifolds belonging to the first two deformation classes, not much is known for the exceptional deformation classes. In this talk I will present joint work with G. Mongardi and A. Rapagnetta, regarding the geometry of O'Grady's six dimensional example. By realizing these examples as "quotients" of another hyperk?hler manifold by a birational involution, we are able to compute the Hodge numbers and, in work in progress, also study properties of their moduli spaces/deformation class.

Consider a smooth projective variety over a number field. The image of the associated (complex) Abel-Jacobi map inside the (transcendental) intermediate Jacobian is a complex abelian variety. We show that this abelian variety admits a distinguished model over the original number field, and use it to address a problem of Mazur on modeling the cohomology of an arbitrary smooth projective variety by that of an abelian variety. (This is joint work with Sebastian Casalaina-Martin and Charles Vial.)

I will report on a joint work with F. Campana. Consider a projective manifold whose canonical bundle is pseudo-effective. We show that the determinant of any quotient of some tensor power of its co-tangent bundle is pseudo-effective as well. This can be seen as a generalization of the important generic semi-positivity result of Y. Miyaoka; some applications will equally be discussed.