Parametric behavior of A-hypergeometric solutions
A-hypergeometric systems are the D-module counterparts of toric ideals, and their behavior is linked closely to the combinatorics of toric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary. In particular, we stratify the parameter space so that solutions are locally analytic within each (connected component of a) stratum. This is joint work with Jens Forsgård and Laura Matusevich.
Lefschetz Properties
The study of Lefschetz properties of artinian algebras has its origin in the Hard Lefschetz Theorem and was encouraged by Stanley’s use of it in the proof of the g-Conjecture for simplicial polytopes. The Weak Lefschetz Property means that multiplication by a general linear form has maximal rank in each degree and the Strong Lefschetz Property that the same also holds for all powers of the form. Despite the fact that most artinian algebras have these properties it is surprisingly hard to prove general results in this direction. I will discuss some recent work together with Migliore, Miró-Roig and Nagel in this area.
Families of ideals with radical gin
We will discuss multigraded ideals with a radical generic initial ideal. We prove that a multigraded ideal has a radical multigraded generic initial ideal then the same is true for every multigraded hyperplane section and for every multigraded projection. Connection to universal Gröbner bases for determinantal ideals, algebras associated to subspaces configurations and multiview varieties will be discussed. We will also present a ``rigidity” conjecture suggested by a theorem of Brion. This is joint work with Emanuela De Negri and Elisa Gorla.
The Prym-Green Conjecture
By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. I will discuss the proof, obtained in joint work with Kemeny, of the Prym-Green Conjecture in odd genus for all levels p. Probabilistic arguments suggested that the conjecture might fail for level 2 and genera with high divisibility property by 2. I will present this phenomenon, focusing on the case of genus 8.
Normal bundles of rational curves in projective space
In the early 1980s, Eisenbud and Van de Ven wrote a pair of papers describing the possible normal bundles of smooth rational curves of degree n in the 3-dimensional projective space. The normal bundle of such a curve necessarily splits as a direct sum of two line bundles, of type (2n-1+a,2n-1-1), and among the results Eisenbud and Van de Ven obtained was that the behavior was, up to a point, exactly what you'd expect based of the deformation theory of such bundles: for a general rational curve, a=0; and in general for a at most n-4 the locus of curves with normal bundle of type (2n-1+a,2n-1-a) is irreducible of the expected codimension 2a-1 in the space of all smooth rational curves.
This raised the question of whether normal bundles of rational curves in higher-dimensional projective spaces behave similarly. The answer, recently obtained by Izzet Coskun and Eric Riedl, is an emphatic ``no:" Coskun and Riedl show that there can be arbitrarily many components in the locus of rational curves with a given normal bundle, and that the dimensions can be arbitrarily greater than the expected. In this talk I'll describe their work and describe other open problems in the area.
What is the maximum projective dimension of n quadrics?
The title question is, of course, a more refined version of Stillman's question. The work of Ananyan and Hochster gave a positive answer to Stillman's question, but effective bounds may be impossible. However, for ideals generated by quadrics there is a guess, and we prove this guess for quadrics of codimension two. This work is joint work with Paolo Mantero, Jason McCullough, and Alexandra Seceleanu.
The journey to this work passes through a conversation with Eisenbud around 1984, many theorems of Eisenbud, and the project started at MSRI!
A local to global principle in modular representation theory
In recent years the local to global principle, which is a basic tool in commutative algebra and algebraic geometry, has been successfully adapted to the context of modular representations of finite groups and group schemes. The aim of my talk will be to explain some of these developments. It is part of joint work with Dave Benson, Henning Krause, and Julia Pevtsova.
Unique factorization in power series rings
We study unique factorization in power series rings over complete, local domains and then apply these results to the moduli of higher dimensional varieties and pairs.
Minimal free resolutions over complete intersections
Hilbert's Syzygy Theorem shows that minimal free resolutions over a polynomial ring are finite. By a result of Serre, it follows that most free resolutions over quotient rings are infinite. Such resolutions can have very intricate structure; for example, Anick constructed minimal free resolutions with irrational generating series. The concept of matrix factorization was introduced by Eisenbud 35 years ago, and it describes completely the asymptotic structure of minimal free resolutions over a hypersurface. The lecture will provide an overview of recent joint work with Eisenbud. We describe the asymptotic structure of minimal free resolutions over complete intersections.
Bertini irreducibility theorems over finite fields
The classical Bertini irreducibility theorem states that if X is a geometrically irreducible subvariety of the projective space over an infinite field k, and the dimension of X is at least 2, then there exists a hyperplane H in the projective space over k whose intersection with X is geometrically irreducible. This can fail if k is finite, but certain variants are true. For instance, we prove that if X is as above but k is finite, then the fraction of degree d hypersurfaces H whose intersection with X is geometrically irreducible tends to 1 as d tends to infinity. This result, which is more difficult than the Bertini smoothness theorem over finite fields proved in 2004, is joint work with François Charles.
Deformation of Canonical morphisms and Moduli spaces
In this talk we will deal with deformations of finite maps and show how to use deformation theory to construct varieties with given invariants in a projective space. Among other things, we give a criterion that determines when a finite map can be deformed to a one-to-one map. We use this general result that holds for all dimensions to construct new surfaces of general type with birational canonical map, for different
and (the canonical map of the surfaces we construct is in fact a finite, birational morphism).
Our results enable us to describe some new components of the moduli of surfaces of general type. We find infinitely many moduli spaces having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree two morphism. We also prove some analogues of these results for varieties of general type in all dimensions. This is joint work with F.J Gallego and M. Gonzalez.
Boij-Söderberg theory for Grassmannians
An explicit description for the cone of Betti tables of graded modules over a polynomial ring was given in work of Eisenbud and Schreyer, and a surprising connection was established with the cone of cohomology tables of vector bundles on projective space. I will explain some ongoing work of Ford and Levinson, some joint with me, which seeks to establish Grassmannian analogues of this where graded modules are replaced by GL_k-equivariant modules for k>1.
Horrocks' splitting on products of projective space
Using the theory of Tate resolutions on products of projective space, I will give a middle cohomology vanishing criterion for a vector bundle to be of the form direct sum of the form O(a_iH) for fixed very ample line divisor H on a product of projective spaces.
Rationality does not specialize among terminal varieties
Hassett, Pirutka, and Tschinkel showed that rationality is not an open condition among smooth complex projective 4-folds. One remaining question is whether rationality is a closed condition in a certain sense. Namely: given a family of smooth projective varieties for which very general fibers are rational, is every fiber rational? We discuss the positive and negative results on this problem if we allow mildly singular (terminal) varieties.
Limits of Hurwitz spaces (in various senses)
I will discuss three projects involving Hurwitz spaces, all with the hope of some sort of structure stabilizing as the genus or gonality becomes large. First (with Matchett Wood, in progress): the class of curves of gonality 3 and 4 (and hopefully 5) in the Grothendieck ring tends to a limit as the genus gets large, analogous to Bhargava's counts of number fields, also in terms of zeta-values. Second (with Patel): the Chow ring of trigonal curves with simple branching is trivial. Third (very much in progress, with Deopurkar and Patel): the class of genus g degree d simply branched covers of the projective line stabilizes as d gets large; the punchline is that one might hope that the limit is a particularly nice answer.
Degenerations of hyper-Kähler manifolds
We prove that if the central fiber of a degeneration of hyper-Kähler manifolds has one component which is not uniruled, then after base-change the family becomes fiberwise birational to a family of smooth hyper-Kähler manifolds. This provides an easy way to compute the deformation types of many hyper-Kähler manifolds. In particular, this gives an easy proof of the fact that the compactified intermediate Jacobian fibrations I constructed with Laza and Saccà are deformation equivalent to O’G10 manifolds. This is joint work with Kollár, Laza and Saccà.
Finite free resolutions and Kac-Moody Lie algebras
In this talk I will discuss the structure of free resolutions of length 3 over Noetherian rings. Associate to a triple of ranks (r_3, r_2, r_1) in our free complex a triple (p,q,r)=(r_3+1, r_2-1, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lengths p-1, q-1, r-1 attached to the central vertex). The main result is the explicit construction of a generic ring R_{gen} for resolutions of the format with the differentials of ranks r_1, r_2, r_3. This ring carries an action of a Kac-Moody Lie algebra associated to the graph T_{p,q,r}. In particular the ring R_{gen} is Noetherian if and only if T_{p,q,r} is a Dynkin graph. I will discuss the structure of the ring R_{gen} and possible consequences for the structure of perfect ideals of codimension 3.