Algebraic Geometry/Commutative Algebra Seminar, 2018–2019

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu.

Spring Schedule

The seminar will meet on Wednesdays, 3:00–4:00pm in 258 Hurley unless otherwise noted. Related events are also listed below.

Fall Schedule

The seminar will meet on Wednesdays, 3:00–4:00pm in 258 Hurley unless otherwise noted. Related events are also listed below.

Abstracts

Aug. 29, 2018

Speaker

Alessio Sammartano (Notre Dame)

Title

Upper bounds for Betti numbers in Hilbert schemes of complete intersections

Abstract

Let S be a polynomial ring over a field and I a homogeneous ideal containing a regular sequence f = f_1, ..., f_c. In this talk we discuss the existence of sharp upper bounds for the syzygies of I in terms of the degrees of the regular sequence f and of the Hilbert function or the Hilbert polynomial of I. This is a joint work with Giulio Caviglia.

Sep. 5, 2018

Speaker

Eric Riedl (Notre Dame)

Title

A Grassmannian technique and the Kobayashi Conjecture

Abstract

An entire curve on a complex variety is a holomorphic map from the complex numbers to the variety. We discuss two well-known conjectures on entire curves on very general high-degree hypersurfaces X in P^n: the Green-Griffiths-Lang Conjecture, which says that the entire curves lie in a proper subvariety of X, and the Kobayashi Conjecture, which says that X contains no entire curves. We prove that (a slightly strengthened version of) the Green-Griffiths-Lang Conjecture in dimension 2n implies the Kobayashi Conjecture in dimension n. Our technique is substantially simpler than previous approaches to this question, and has already led to improved bounds for the Kobayashi Conjecture. This is joint work with David Yang.

Sep. 12, 2018

Speaker

Michael Wibmer (Notre Dame)

Title

Difference algebraic groups

Abstract

Difference algebraic groups are subgroups of the general linear group defined by algebraic difference equations in the matrix entries. The theory of these groups has some similarities with the theory of linear algebraic groups. We will discuss some finiteness properties of difference algebraic groups and applications to differential equations.

Sep. 19, 2018

Speaker

David Hansen (Notre Dame)

Title

The one-point compactification of a scheme

Abstract

The one-point compactification of a locally compact Hausdorff space is a classical and useful construction. In this talk I'll define an analogous compactification in the setting of schemes, explain its basic properties, and give some applications. In particular, I will sketch a totally canonical definition of the functor Rf_! in etale cohomology. This is joint work in progress with Johan de Jong.

Sep. 26, 2018

Speaker

Whitney Liske (Notre Dame)

Title

Defining equations of the Rees algebra for a family of ideals

Abstract

Let $R=k[x_1, \ldots, x_d]$ be a polynomial ring in $d$ variables over a field $k$. Let $m =(x_1, \ldots, x_d)$ be the maximal homogenous ideal of $R$. Let $I$ be a Gorenstein ideal generated by all the generators of $m^2$ except for one. For each fixed $d$ these ideals are all equivalent, up to change of coordinates. The goal is to compute the defining equations of the special fiber ring and the Rees ring of these ideals. To compute the Rees ring, we study the Jacobian dual and the defining equations of the special fiber ring of $m^2$.

Oct. 5, 2018

Speaker

David Harbater (University of Pennsylvania)

Title

Group realization and embedding problems in differential Galois theory

Abstract

Much of the progress in Galois theory in recent decades has concerned function fields of curves, with results obtained about realizing finite groups as Galois groups, solving Galois embedding problems, and understanding the structure of absolute Galois groups in that context. More recently, analogous results have been obtained in differential Galois theory, in part by the use of related methods, and this talk will discuss such results. This work is joint with Annette Bachmayr, Julia Hartmann, Florian Pop, and Michael Wibmer.

Oct. 12, 2018

Speaker

Kevin Tucker (UIC)

Title

Symbolic and Ordinary Powers of Ideals in Hibi Rings

Abstract

The study of symbolic powers has a long history in commutative algebra and algebraic geometry. Celebrated results of Ein-Lazarsfeld-Smith, Hochster-Huneke, and recently Schwede-Ma give uniform comparison theorems for symbolic and ordinary powers of prime ideals in regular rings. In this talk, I will review some of these results, with a view towards recent work on extending such bounds to certain singular rings. In particular, we discuss such bounds for a class of rings in combinatorial commutative algebra called Hibi rings. Roughly speaking, a Hibi ring is the affine toric ring associated to the order polytope of a finite poset. In joint work with Page and Smolkin, we show that many Hibi rings satisfy precisely the same uniform symbolic power bounds as regular rings, and moreover we give limited symbolic power bounds for all Hibi rings.

Oct. 24, 2018

Speaker

Claudiu Raicu (Notre Dame)

Title

Koszul Modules and Green’s Conjecture

Abstract

Formulated in 1984, Green’s Conjecture predicts that one can recognize the intrinsic complexity of a smooth algebraic curve from the syzygies of its canonical embedding. In characteristic zero, Green’s Conjecture for a general curve has been resolved using geometric methods in two landmark papers by Voisin in the early 00s. More direct approaches have been proposed over the years to solve Green’s Conjecture for general curves, and one dates back to a paper of Eisenbud in the early 90s, and involves a connection with the syzygies of the tangent developable T to a rational normal curve. I will explain how the theory of Koszul modules allows for a complete characterization, in arbitrary characteristics, of the (non-)vanishing behavior of the syzygies of T, proving Green’s conjecture for general curves in almost all characteristics. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.

Oct. 31, 2018

Speaker

Marius Vlădoiu (University of Bucharest / Purdue)

Title

Hypergraph encodings of arbitrary toric ideals

Abstract

We discuss a combinatorial classification of all toric ideals, via the bouquet structure, with several consequences on some open questions. We also show that hypergraphs exhibit a surprisingly general behavior: the toric ideal associated to any general matrix can be encoded by that of a 0/1 matrix, while preserving the essential combinatorics of the original ideal. Furthermore we provide a polarization-type operation for arbitrary positively graded toric ideals, which preserves all the combinatorial signatures and the homological properties of the original toric ideal. This talk is based on joint works with Sonja Petrovi\'c and Apostolos Thoma.

Nov. 7, 2018

Speaker

Andrei Jorza (Notre Dame)

Title

The Witten zeta function of projective varieties

Abstract

In the 80's Witten computed the volumes of certain moduli spaces of flat connections on a compact Riemann surface in terms of the special values of the Witten zeta function $\zeta_G(s)$, a Dirichlet series attached to a complex semisimple group $G$. Ten years ago Larsen and Lubotzky computed the abscissa of convergence of $\zeta_G(s)$ as the ratio of the rank of $G$ by the number of positive roots. In ongoing work with Benjamin Bakker we introduced a Witten zeta function $\zeta_X(s)$ for a complex projective variety $X$. We showed that its abscissa of convergence can often be expressed as a ratio of the Picard rank by the dimension of a variety $Y$ (not necessarily $X$) and found combinatorial bounds on the growth rate at the pole. In separate recent work with Sam Evens we showed that when $X$ is a flag variety or the wonderful compactification of a split de Concini-Procesi pair then $\zeta_X(s)$ admits meromorphic continuation with simple pole to a neighborhood of the abscissa of convergence. In the special case of SL(3) we compute the residue using a formula of Zagier on multiple zeta functions.

Nov. 14, 2018

Speaker

Howard Nuer (UIC)

Title

MMP and wall-crossing for Bridgeland moduli spaces on Enriques and bielliptic surfaces

Abstract

Since Bridgeland introduced his mathematical formulation of Douglas’s pi-stability, Bridgeland stability conditions have become a powerful tool for answering many questions in the study of coherent sheaves on varieties, especially with regard to the birational geometry of their moduli. In this talk, I will report on the application of this perspective to the study of stable sheaves on Enriques and bielliptic surfaces. In joint work with K. Yoshioka, we prove that any two moduli spaces of Bridgeland stable objects of Mukai vector v with respect to two generic stability conditions are birational. We achieve this by completely classifying the geometric behavior induced by crossing any given wall W. We further conjecture that all minimal models of these (often singular) moduli spaces arise as Bridgeland moduli. In solo authored work, I obtain a similar classification for bielliptic surfaces, proving on the way many heretofore unknown fundamental results about moduli of sheaves and Bridgeland stable objects on bielliptic surfaces (such as the existence of coarse projective Bridgeland moduli spaces and criteria for their nonemptiness).

Nov. 28, 2018

Speaker

Izzet Coskun (UIC)

Title

Brill-Noether Theorems for sheaves on surfaces

Abstract

I will discuss joint work with Jack Huizenga on the cohomology of the general stable sheaf on a rational surface. We determine the cohomology of the general stable sheaf on Hirzebruch surfaces. As a consequence, we classify the Chern characters for which the general stable sheaf is globally generated. These theorems have many applications. For example, we prove sharp Bogomolov inequalities on Hirzebruch surfaces for any polarization and obtain a classification of stable Chern characters. If time permits, I will describe analogous results with Howard Nuer and Kota Yoshioka on the cohomology of the general stable sheaf on K3 surfaces.

Dec. 5, 2018

Speaker

Ritvik Ramkumar (U.C. Berkeley)

Title

The Hilbert scheme of a pair of linear spaces

Abstract

The Grassmannian is a smooth moduli space with very rich geometry that parameterizes simple varieties, namely linear spaces. One can study a "natural" generalization, the component of a Hilbert scheme that parameterizes a pair of linear spaces in $\mathbb P^n$. In this talk we will describe a rigidity result that allows us to completely control degenerations in this component. We will then use it to give new examples of smooth components and describe them as blowups of certain (products) of Grassmannians. Time permitting, we will also describe the singularities of other components meeting these smooth components.

Feb. 6, 2019

Speaker

Anand Pillay (Notre Dame)

Title

Open subgroups of p-adic algebraic groups

Abstract

I discuss the problem of whether open subgroups of p-adic algebraic groups are "p-adic semialgebraic". The corresponding fact is true in the real case. I will give definitions, background, and motivation.

Feb. 20, 2019

Speaker

Matthew Dyer (Notre Dame)

Title

Variations on independence of characters

Abstract

We discuss extensions of Dedekind’s lemma on linear independence of characters, and related complements involving Galois connections and adjoint functors, together with some applications and open questions.

Feb. 27, 2019

Speaker

Jacob Matherne (IAS)

Title

Kazhdan-Lusztig polynomials of matroids

Abstract

Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, attracting a great deal of research in geometric representation theory, and providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids. I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.

Mar. 6, 2019

Speaker

Juan Migliore (Notre Dame)

Title

Unexpected Hypersurfaces

Abstract

Given a linear system of hypersurfaces in projective space (for example), there are various ways of imposing geometric constraints on it, and in general for each such constraint there is an expected number of conditions imposed on the dimension of the linear system. It is interesting to try to understand what kind of constraints, and what kind of geometric properties of the linear system, can result in the imposition of fewer than the expected number of conditions. This leads to the notion of unexpected hypersurfaces. I’ll talk about some recent results in this direction.

Mar. 20, 2019

Speaker

Shizhang Li (Columbia)

Title

An example of liftings with different Hodge numbers

Abstract

Does a smooth proper variety in positive characteristic know the Hodge number of its liftings? The answer is ”of course not”. However, it’s not that easy to come up with a counter-example. In this talk, I will first introduce the background of this problem. Then I shall discuss some obvious constraints of constructing a counter-example. Lastly I will present such a counter-example and state a further question.

Mar. 27, 2019

Speaker

Wenliang Zhang (UIC)

Title

An analogue of the Hartshorne-Polini theorem in positive characteristic

Abstract

Recently, Hartshorne and Polini proved a theorem to characterize the dimension of certain de Rham cohomology groups of a holonomic D-module over complex numbers as the number of specific D-linear maps associated with the holonomic D-module. In this talk, I will explain an analogue of this result in positive characteristic for F-finite F-modules. This is a joint work with Nicholas Switala.

Apr. 5, 2019

Speaker

Alex Suciu (Northeastern)

Title

Duality, finiteness, and cohomology jump loci

Abstract

A recurring theme in topology is to determine the duality and finiteness properties of spaces and groups. I will discuss some of the interplay between these properties, the structure of algebraic models associated to them, and the geometry of the corresponding cohomology jump loci. Furthermore, I will outline some of the applications of this theory to complex algebraic geometry and low-dimensional topology.

Apr. 17, 2019

Speaker

Amy Huang (Wisconsin)

Title

Syzygies of Determinantal Thickenings via General Linear Lie Superalgebra Representations

Abstract

The coordinate ring $S = \mathbb{C}[x_{i,j}]$ of space of $m \times n$ matrices carries an action of the group $\mathrm{GL}_m \times \mathrm{GL}_n$ via row and column operations on the matrix entries. If we consider any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$ in $S$, the syzygy modules $\mathrm{Tor}_i(I,\mathbb{C})$ will carry a natural action of $\mathrm{GL}_m \times \mathrm{GL}_n$. Via BGG correspondence, they also carry an action of $\bigwedge^{\bullet} (\mathbb{C}^m \otimes \mathbb{C}^n)$. It is a recent result by Raicu and Weyman that we can combine these actions together and make them modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We will explain how this works and how it enables us to compute all Betti numbers of any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$. The latter part will involve combinatorics of Dyck paths.

Apr. 24, 2019

Speaker

Daniele Rosso (Indiana University Northwest)

Title

Twisted generalized Weyl algebras over polynomial rings

Abstract

Twisted generalized Weyl algebras (TGWAs) are a family of algebras that includes as special cases a lot of algebras of interest in representation theory. They are defined by generators and relations starting from a base ring R and the choice of an n-tuple of elements in the center of R, together with an n-tuple of commuting automorphisms of R, that have to satisfy certain consistency equations. I will explain how to classify TGWAs, up to graded isomorphism, in the case where R is a polynomial ring over a field of characteristic zero. This is joint work with Jonas Hartwig.

May 1, 2019

Speaker

Yajnaseni Dutta (Northwestern)

Title

Fujita type conjectures for pushforwards of pluricanonical sheaves

Abstract

Extending the property that a line bundle on a smooth projective curve is globally generated if its degree is bigger than 2g, Takao Fujita, in 1985, conjectured that there is an effective bound on the twists by an ample line bundle to obtain global generation for canonical bundles. Even though the conjecture remains unsolved as of today, based on Demailly's singular divisor techniques, partial progress was made by Angehrn-Siu, Ein-Lazarsfeld, Heier, Helmke, Kawamata, Reider, Ye-Zhu et al. In this talk I will focus on similar global generation conjecture due to Popa and Schnell for pushforwards of canonical and pluricanonical bundles under certain morphisms f: Y --> X. The canonical bundle case first appeared in Kawamata's work in 2002 and the proof used Hodge theoretic techniques combined with the Demaiily's singularity techniques. In this talk I will present a generic global generation result for log canonical pairs building on Kawamata's theorem. I will also discuss weak positivity properties of these pushforwards and its implications toward subadditivity of Kodaira dimensions. Some parts of this work was done jointly with Takumi Murayama.