Schedule of talks:

See also the Arithmetic Geometry Seminar

Abstracts

(Anderson): Using the geometry of certain infinite-dimensional flag varieties, I will describe a variation on the back-stable Schubert polynomials introduced by Lam, Lee, and Shimozono. New morphisms among these varieties become apparent in the infinite limit, and they imply certain properties of the corresponding Schubert polynomials. The direct sum morphism plays an especially interesting role. This is partially based on joint work with William Fulton.

(Huizenga): In algebraic geometry, the vector bundles on an algebraic variety form continuous families known as moduli spaces. Vector bundles corresponding to special points in the moduli space often have bad properties, but the vector bundles corresponding to general points in the moduli space are often better behaved. For instance, the general vector bundle might have cohomology groups which are easy to describe. One of the most common ways of describing geometric loci in the moduli space is to study the locus where cohomology jumps, or the locus where the cohomology of a tensor product jumps. These loci are the source of many interesting subvarieties and control the birational geometry of the moduli spaces. In this talk I will discuss recent work which computes the cohomology of a general tensor product of vector bundles on the projective plane. This is joint work with Izzet Coskun and John Kopper.

(Ji): The classical Noether-Lefschetz theorem says that for a very general surface S of degree 4 in P³ over the complex numbers, the restriction map from the divisor class group on P³ to S is an isomorphism. In this talk, we will show a Noether-Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

(Usatine): Matroids are versatile combinatorial objects with deep connections to many subjects, including chromatic polynomials of graphs, arrangements of vectors and hyperplanes, and the geometry of Grassmannians. I will discuss joint work with Dhruv Ranganathan in which we use Gromov-Witten theory to associate new invariants to matroids. In particular, I will explain how to obtain a canonical (independent of chosen realization) quantum deformation of any realizable matroid's cohomology ring.

(Hong): By a theorem of Evans-Mirkovic, the smooth locus of a spherical Schubert variety in affine Grassmannian is the big Schubert cell. One may ask the similar question for Schubert varieties in twisted affine Grassmannian. It was conjectured by Haines-Richarz that similar result should be true for absolutely special types of twisted affine Grassmannian. In this talk, I will explain a proof of this conjecture. We use Zhu's methods and results on the duality between Demazure modules and torus fixed-point subscheme of Schubert varieties. We also use global Schubert variety of parahoric Bruhat-Tits group scheme, which relates twisted and untwisted Schubert varieties. This talk is based on the joint work with Marc Besson.

(Doran): I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Alan Thompson (arXiv:2109.04849).

(Ullery): If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo's Lemma about points on rational curves. I will then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.

(Sutherland): Given a general polynomial f_n(z) of degree n, how can we (algebraically) determine a root of f_n(z) in terms of its coefficients as simply as possible? Resolvent degree is an invariant measuring complexity that is both classically motivated by this problem and widely applicable, as we can study the resolvent degree of branched covers of varieties, field extensions, groups, representations, etc. In this talk, we will introduce resolvent degree and examine recent upper bounds on the resolvent degree of solving general polynomials, which are obtained by determining special points on certain varieties. Some of the results in this talk are joint with Curtis Heberle.

(Cavey): The Hilbert scheme of n points in the plane parametrizes finite, length n subschemes of C². In this talk I will explain how to compute the Newton-Okounkov body of this Hilbert scheme. Newton-Okounkov bodies are convex sets that encode geometric information about the underlying space. In this case, the Newton-Okounkov body turns out to be an (unbounded) polyhedron which we can describe explicitly. If time permits, I will also discuss partial results for the Hilbert schemes of points on toric surfaces.

(Gallardo): We report ongoing work with Benjamin Schmidt on compactifications of the moduli of unlabelled points in the plane. These compactifications are obtained via GIT quotients of both the Hilbert scheme and its birational models constructed with Bridgeland stability. In particular, we describe the role of non-reduced points and the existence of a compactification that is the opposite of the familiar Chow quotient.

(Dumitrescu): In this talk, we construct examples of rigid curves in Pⁿ that we call (-1) curves. We investigate moving curves in Pⁿ that we call (0) and (1) curves and we explore their applications. We further construct a stratification of the Mori cone of curves encoding the birational geometry of the ambient space.

(Castor): In this talk, we will discuss methods for studying singularities on projective hypersurfaces. We first provide background necessary for understanding monodromy and spectra. We then compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on a projective hypersurface of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound. We then provide a purely combinatorial interpretation of our theorems and our conjecture.

(Zhang): The Abel-Jacobi map on a smooth projective curve is a group homomorphism which sends divisors of degree zero to the Jacobian of the curve. In fact, the Abel-Jacobi image can be also extracted from the mixed Hodge structure on the first cohomology of the complement of the support of thedivisor. Similar argument holds in higher dimensions for Griffiths' Abel-Jacobi map. In 2015, Xiaolei Zhao defined a notion called Topological Abel-Jacobi map, which is a generalization of Griffiths' Abel-Jacobi map to topological cycles. We will show it coincides with an alternative definition using R-splitting properties of certain mixed Hodge structures suggested by Christian Schnell.

Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2020-2021

Ohio State University Algebraic Geometry Seminar-Year 2019-2020

Ohio State University Algebraic Geometry Seminar-Year 2018-2019

Ohio State University Algebraic Geometry Seminar-Year 2017-2018

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

This page is maintained by Angie Cueto and Dave Anderson.