Algebra and Algebraic Geometry Seminar Spring 2023

The seminar normally meets 2:30-3:30pm on Fridays, in the room VV B223.

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Spring 2023 Schedule

date	speaker	title	host/link to talk
February 3	Dima Arinkin	Integrating symplectic stacks	Local
February 24	Ruijie Yang	Higher multiplier ideals	Maxim
March 10	Joerg Schuermann	Equivariant motivic characteristic classes of Schubert cells	Maxim
March 24	Daniel Corey	Degenerations of flag varieties and subdivisions of generalized permutahedra	Rodriguez
March 31	Tudor Padurariu	Categorical and K-theoretic Donaldson-Thomas theory of C^3	Maxim
April 14	Lena Ji	Rationality of conic bundle threefolds over non-closed field	Rodriguez/Ellenberg
April 21	Christopher O'Neill	Numerical semigroups, minimal presentations, and posets	Sobieska
April 28	Ayah Almousa	Gröbner basis techniques for determinantal facet ideals	Rodriguez
May 5	James Hotchkiss	The period-index problem over the complex numbers	Caldararu
May 12	Yash Deshmukh	A homotopical description of Deligne-Mumford compactifications	Caldararu

Abstracts

Dima Arinkin

Integrating symplectic stacks

Shifted symplectic stacks, introduced by Pantev, Toën, Vaquie, and Vezzosi, are a natural generalization of symplectic manifolds in derived algebraic geometry. The word `shifted' here refers to cohomological shift, which can naturally occur in the derived setting: after all, the tangent space is now not a vector space, but a complex. Several classes of interesting moduli stacks carry shifted simplectic structures.

In my talk (based on a joint project with T.Pantev and B.Toën), I will present a way to generate shifted symplectic stacks. Informally, it involves integration along a (compact oriented) topological manifold X: starting with a family of shifted symplectic stacks over X, we produce a new stack of sections of this family, and equip it with a symplectic structure via an appropriate version of the Poincaré duality.

Ruijie Yang

Higher multiplier ideals

For any Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many important applications in algebraic geometry, complex analytic geometry and commutative algebra. It turns out that this is only a small piece of a larger picture. In this talk, I will discuss the construction of a family of ideal sheaves associated to (X,D), indexed by an integer indicating the Hodge level, such that the lowest level recovers the usual multiplier ideals. We study their local and global properties systematically: the local properties rely on Saito's theory of mixed Hodge modules and some results inspired by Sabbah's theory of twistor D-modules; while the global properties need Sabbah-Schnell's theory of complex mixed Hodge modules and Beilinson-Bernstein’s theory of twisted D-modules from geometric representation theory. I will also compare this with the theory of Hodge ideals recently developed by Mustata and Popa. If time permits, I will discuss some application to the Riemann-Schottky problem via the singularity of theta divisors on principally polarized abelian varieties. This is based on the joint work in progress with Christian Schnell. I may also discuss some application to a homological characterization of higher rational and higher Du Bois singularities, based on joint work with Laurentiu Maxim.

Joerg Schuermann

Equivariant motivic characteristic classes of Schubert cells

We explain in the context of complete flag varieties X=G/B the inductive calculation of equivariant motivic characteristic classes of Schubert cells via suitable Demazure-Lusztig operators, fitting with convolution actions of corresponding Hecke-algebras and Weyl groups. This is joint work with P. Aluffi, L. Mihalcea and C. Su.

Daniel Corey

Degenerations of flag varieties and subdivisions of generalized permutahedra

We study the initial degenerations of the type-A flag varieties, and show how they are related to flag matroid strata and subdivisions of flag matroid polytopes. As applications, we give a complete proof of a conjecture of Keel and Tevelev on log canonical compactifications of moduli spaces of hyperplanes in projective space in general position, and study the Chow quotient of the complete flag variety by the diagonal torus of the projective linear group. This is based on joint work with Olarte and Luber.

Tudor Padurariu

Categorical and K-theoretic Donaldson-Thomas theory of C^3

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is p(d), the number of plane partitions of d. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about categorical and K-theoretic refinements of DT invariants, focusing on the explicit case of C^3. In particular, we show that the K-theoretic DT invariant for d points on C^3 also equals p(d). This is joint work with Yukinobu Toda.

Lena Ji

Rationality of conic bundle threefolds over non-closed field

The intermediate Jacobian obstruction to rationality for complex threefolds was introduced by Clemens--Griffiths in their proof of the irrationality of the cubic threefold. For conic bundles over P^2, this obstruction characterizes rationality over the complex numbers. Recently, over non-closed fields k, Hassett--Tschinkel and Benoist--Wittenberg refined this obstruction by defining torsors over the intermediate Jacobian. For Fano threefolds of Picard rank 1, this refined obstruction can be used to characterize k-rationality. In this talk, we study the IJ torsor obstruction for conic bundles and explain why it does not characterize k-rationality in this higher Picard rank setting. This talk is based on joint work with S. Frei--S. Sankar--B. Viray--I. Vogt and joint work with M. Ji.

Christopher O'Neill

Numerical semigroups, minimal presentations, and posets

A numerical semigroup is a subset S of the natural numbers that is closed under addition.  One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a minimal binomial generating set for the defining toric ideal of S).  In this talk, we present a method of constructing a minimal presentation of S from a portion of its divisibility poset. Time permitting, we will explore connections to polyhedral geometry.  

No familiarity with numerical semigroups or toric ideals will be assumed for this talk.  

Ayah Almousa

Gröbner basis techniques for determinantal facet ideals

The ideal of maximal minors of a generic matrix possesses many surprising and desirable properties that make them particularly tractable to study using Gröbner basis techniques. For example, Sturmfels and Zelevinsky showed that the natural generating set consisting of all maximal minors forms a reduced Gröbner basis of the ideal with respect to any monomial order. Boocher went one step further and showed that the graded Betti numbers of an ideal of maximal minors and any of its initial ideals must also agree. In addition, Conca, Herzog, and Valla showed that one can compute the defining ideals of the Rees algebra and special fiber ring of the ideal of maximal minors using SAGBI ("Subalgebra Analogue to Gröbner Bases for Ideals") bases; this means that one can understand the relations between the subalgebra generated by maximal minors of a generic matrix by first studying the relations by the subalgebra generated by initial terms of the maximal minors with respect to some monomial ordering.

One direction for generalizing ideals of maximal minors is to imagine that the column-sets of minors appearing in the generating set are parametrized by some simplicial complex. Such ideals are called determinantal facet ideals (DFIs), and their study was introduced by Ene, Herzog, and Hibi. In the case that when the simplicial complex is a graph, DFIs are better known as binomial edge ideals, and they are even more well behaved than arbitrary DFIs: many authors have shown that invariants of the graph reflect or bound homological invariants of the corresponding ideal. The study of DFIs turns out to be much more subtle than the study of the ideal of all maximal minors of a generic matrix and has seen comparably less attention, even though such ideals and their primary decompositions arise naturally as conditional independent statement ideals in algebraic statistics by work of Herzog-Hibi-Hreinsdóttir-Kahle-Rauh.

In this talk, we will give an overview of what is known about DFIs and discuss some open questions. The discussion will center on applications of Gröbner basis and SAGBI basis techniques for computing Rees algebras, syzygies, and graded Betti numbers of large classes of these ideals. Some of this is joint work with Keller VandeBogert, and some of this is joint work with Kuei-Nuan Lin and Whitney Liske.

James Hotchkiss

The period-index problem over the complex numbers

The period-index problem is a longstanding question about the complexity of Brauer classes over a field. I will discuss some Hodge-theoretic aspects of the problem for complex function fields, and give some applications to Brauer groups and the integral Hodge conjecture.

Yash Deshmukh

A homotopical description of Deligne-Mumford compactifications

I will explain how the Deligne-Mumford compactifications of moduli spaces of curves (of all genera) arise from the moduli spaces of framed curves by homotopically trivializing certain circle actions in an appropriate sense. I will sketch how this is relevant to the problem of relating GW invariants (in all genera) with Fukaya categories. Finally, I will indicate how our result relates to other statements available in the literature.