Algebraic Geometry/Commutative Algebra Seminar, 2021–2022

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

Spring Schedule

The seminar will meet on Tuesdays, 2:30-3:30pm, either on Zoom or in 258 Hurley, unless otherwise noted. Related events are also listed below.

Fall Schedule

The seminar will meet on Tuesdays, 2:30–3:30pm, either on Zoom or in Hayes-Healy 125, unless otherwise noted. Related events are also listed below.

Abstracts

Aug. 31, 2021

Speaker

Zhao Gao (Notre Dame)

Title

Cohomology of line bundles on the incidence correspondence

Abstract

For a finite dimensional vector space $V$, we consider the incidence correspondence $X\subset\mathbb{P}V \times \mathbb{P}V^{\vee}$. We completely characterized the vanishing and non-vanishing of the cohomology groups of line bundles on $X$ in characteristic $p>0$. If $\dim V=3$, this is the result of Griffith from the 70s. In characteristic $0$ case, the cohomology groups are described for all $V$ by the Borel-Weil-Bott theorem. In this talk, we will give graphical description of the nonvanishing behavior and sketch of proof. If time permits, we will investigate the character formula of the cohomology groups. This is joint work with Claudiu Raicu.

Sep. 7, 2021

Speaker

Evan O'Dorney (Notre Dame)

Title

Reflection theorems for number rings

Abstract

Scholz's celebrated 1932 reflection principle, relating the 3-torsion in the class groups of Q(√D) and Q(√-3D), can be viewed as an equality among the numbers of cubic fields of different discriminants. In 1997, Y. Ohno discovered (quite by accident) a beautiful reflection identity relating the number of cubic rings, equivalently binary cubic forms, of discriminants D and -27D, where D is not necessarily squarefree. This was proved in 1998 by Nakagawa, but the proof is rather opaque. In my talk, I will present a new and more illuminating method for proving identities of this type, based on Poisson summation on adelic cohomology (in the style of Tate's thesis). I have found extensions to quadratic forms and quartic forms and rings and to the function-field setting, where they relate threefold (possibly singular) covers of a curve.

Sep. 14, 2021

Speaker

Keller VandeBogert (Notre Dame)

Title

Iterated Mapping Cones for Strongly Koszul Algebras

Abstract

In the study of monomial ideals in polynomial rings, one method of constructing free resolutions if via iterated mapping cones. If the ideal under consideration is well behaved (ie, has linear quotients), then the resulting resolution may also be minimal. This was used by Herzog and Takayama to construct a minimal free resolution for certain classes of ideals admitting linear quotients which generalized other resolutions appearing in the literature, such as Eliahou-Kervaire. In this talk, we will see the extent to which iterated mapping cones for monomial ideals can be extended to Koszul algebras, including some of the difficulties in doing so, and construct a generalized version of the Herzog-Takayama resolution.

Sep. 21, 2021

Speaker

Ritvik Ramkumar (Berkeley)

Title

Rational singularities of nested Hilbert schemes

Abstract

For a smooth surface S the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano.

Sep. 28, 2021

Speaker

Matthew Weaver (Purdue)

Title

Rees Algebras of Codimension Three Gorenstein Ideals of Hypersurface Rings and their Defining Equations

Abstract

One of the most natural ways to study the Rees algebra of ideal is through its defining ideal and its generators, the defining equations. Unfortunately determining such a minimal generating set is difficult in general and results are only known for Rees algebras of specific classes of ideals. In particular, the Rees algebra of a perfect Gorenstein ideal of codimension three has been studied extensively in recent years, but only when such an ideal belongs to a polynomial ring. In this talk we extend some of these results to the situation of the Rees algebra of such an ideal of a hypersurface ring and explore the defining equations. By introducing the modified Jacobian dual and a recursive algorithm of gcd-iterations we produce a minimal generating set of the defining ideal and determine the Cohen-Macaulayness of the Rees algebra.

Oct. 5, 2021

Speaker

Andres Fernandez Herrero (Cornell)

Title

Moduli of sheaves via affine Grassmannians

Abstract

A useful tool in the study of the moduli space of stable vector bundles on a smooth curve C is the existence of the Mumford compactification, which is constructed by adding a boundary parametrizing semistable vector bundles. If the smooth curve C is replaced by a higher dimensional variety X, then one can compactify the moduli problem by allowing vector bundles to degenerate to an object known as a "torsion-free sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsion-free sheaves on such a variety X. More generally, Simpson proved the existence of moduli spaces of semistable pure sheaves supported on smaller subvarieties of X. All of these constructions use the methods of geometric invariant theory (GIT).

The moduli problem of sheaves on X is more naturally parametrized by a geometric object M called an "algebraic stack". In this talk I will explain an alternative GIT-free construction of the moduli space of semistable pure sheaves that is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of $\Theta$-stability introduced by Halpern-Leistner and some recent methods developed by Alper, Halpern-Leistner and Heinloth. In order to apply these recent results, one needs to show some monotonicity conditions for a polynomial numerical invariant on the stack. We prove monotonicity by defining a higher dimensional analogue of the affine Grassmannian for pure sheaves. If time allows, I will also explain how these ideas can be applied to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.

Oct. 12, 2021

Speaker

Jenna Tarasova (Purdue)

Title

Residual Intersections of Determinantal Ideals of $2\times n$ Matrices

Abstract

In this talk we prove that $n$-residual intersections of ideals generated by $2\times 2$ minors of generic $2\times n$ matrices can be written as a sum of links.

Oct. 26, 2021

Speaker

Geoffrey Smith (UIC)

Title

Very free rational curves in Fano varieties

Abstract

I will present a result allowing us to control the normal bundle of a rational curve in certain complete intersections in a variety X. In particular, given a rational curve C in X, under certain hypotheses this control allows us to find complete intersections in X such that the normal bundle to C in Y is "as general as possible." By using this tool, I will present some new examples of separably rational connected Fano varieties in arbitrary characteristic. For instance, a general Fano complete intersection of hypersurfaces of degree at least 3 in a Grassmannian is separably rationally connected in any characteristic. This talk is based on joint work with Izzet Coskun.

Oct. 28, 2021

Speaker

Kevin Tucker (UIC)

Title

Splinter rings and Global +-regularity

Abstract

A Noetherian ring is a splinter if it is a direct summand of every finite cover. Perhaps owing to their simple definition, basic questions about splinters are often devilishly difficult to answer. For example, Hochster's direct summand conjecture is the modest assertion that a regular ring of any characteristic is a splinter, and was finally settled by André in mixed characteristic more than three decades after Hochster's verification of the equal characteristic case using Frobenius techniques. In this talk, I will discuss some recent work on splinter rings in both positive and mixed characteristics. In particular, inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This is based on joint works arXiv:2103.10525 with Rankeya Datta and arXiv:2012.15801 with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Joe Waldron, and Jakub Witaszek.

Nov. 2, 2021

Speaker

Hang Huang (Texas A&M)

Title

The concept of geproci subsets of projective 3-space

Abstract

Tensors are multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. This talk assumes no background in geometry or algebra.

Nov. 9, 2021

Speaker

Brian Harbourne (Nebraska)

Title

The concept of geproci subsets of projective 3-space

Abstract

The occurrence of finite subsets Z in projective 3-space whose general projection to the projective plane is a complete intersection was raised in 2011 by F. Polizzi. Such sets are now called geproci sets. One example is given by a complete intersection in a plane. Another is given by a grid of lines on a smooth quadric. The fact that there are other examples became known only in 2018 as a by-product of work on unexpected surfaces, which in turn was motivated by work on hyperplane arrangements. I will review how these concepts are related and discuss recent results on geproci sets.

Nov. 16, 2021

Speaker

Jonathan Montaño (New Mexico State)

Title

Blowup algebras of determinantal ideals in prime characteristic

Abstract

We study F-purity and strong F-regularity of blowup algebras. Our main focus is on algebras given by symbolic and ordinary powers of different types of determinantal ideals. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their projective dimension stabilizes. To obtain these results we develop the notion of F-pure filtrations and symbolic F-purity. This is joint work with Alessandro De Stefani and Luis Núñez-Betancourt.

Nov. 16, 2021

Speaker

Qaasim Shafi (Imperial College)

Title

Gromov-Witten Invariants of Blow-Ups

Abstract

Gromov-Witten invariants play an essential role in mirror symmetry and enumerative geometry. Despite this, there are few effective tools for computing Gromov-Witten invariants of blow-ups. Blow-ups of X can be rewritten as subvarieties of Grassmann bundles over X. In joint work with Tom Coates and Wendelin Lutz, we exploit this fact and extend the abelian/non-abelian correspondence, a modern tool in Gromov-Witten theory. Combining these two steps allows us to get at the genus 0 invariants of a large class of blow-ups.

Nov. 23, 2021

Speaker

Gregory Taylor (UIC)

Title

Asymptotic syzygies of secant varieties of curves

Abstract

In this talk, we will discuss the asymptotic behavior of the minimal free resolution of the secant variety of a smooth curve. In particular, we will cover the asymptotic purity of the Boij-Soederberg decomposition, some of its corollaries, and directions for further inquiry.

Nov. 30, 2021

Speaker

Richard Birkett (Notre Dame)

Title

Dynamically Stabilising Birational Surface Maps: Two Methods

Abstract

We provide two new approaches to a theorem of Diller and Favre. Namely, a birational self-map $f : X \dashrightarrow X$ on a smooth projective surface $X$ is birationally conjugate to a map which is algebraically stable. Among other things, the first approach confirms the validity of a longstanding practical method to stabilise birational maps. Secondly we show that a map becomes algebraically stable if one repeatedly lifts the map to its graph.

Dec. 7, 2021

Speaker

Emanuela Marangone (Notre Dame)

Title

The non-Lefschetz locus for vector bundle of rank $2$ on $\mathbb{P}^2$

Abstract

A finite length graded $R$-module $M$ has the Weak Lefschetz Property if there is a linear element $\ell$ in $R$ such that the multiplication map $\times\ell: M_i\to M_{i+1}$ has maximal rank. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. I focus on the study of the non-Lefschetz locus for the first cohomology module $H_*^1(\mathbb{P}^2,\mathcal{E})$ of a locally free sheaf $\mathcal{E}$ of rank $2$ over $\mathbb{P}^2$. The main result is that this non-Lefschetz locus has the expected codimension under the assumption that $\mathcal{E}$ is general.

Jan. 25, 2022

Speaker

Boming Jia (Chicago)

Title

Geometry of the Affine Closure of T^*(SL_n/U)

Abstract

In this talk, we will discuss geometric properties of the affine closure of the cotangent bundle T^*(G/U). We will consider the case G=SL_n, and show that $\overline{T^*(SL_n/U)}$ has symplectic singularity (in the sense of Beauville). A double quiver construction of this affine closure by Dancer, Kirwan, Swann will be explained. In particular, when n=3, we can use this construction to show that this affine closure $\overline{T^*(SL_3/U)}$ is isomorphic to the closure of the minimal nilpotent orbit in so(8,C). Moreover, the quasi-classical Gelfand-Graev action constructed by Ginzburg and Kazhdan, can be identified with the restriction of the triality action on so(8) to the closure of the minimal orbit.

Feb. 8, 2022

Speaker

Hang Huang (Texas A&M)

Title

Tensor Ranks and Matrix Multiplication Complexity

Abstract

Tensors can be thought of as multi-dimensional arrays. Many problems in data analysis, physics, and other areas of science can be reformulated as finding rank or border rank decompositions of certain tensors. Notions of ranks and border rank also abound in the literature. In this talk, I will give an overview of the current state of the art of techniques computing different notions of tensor ranks and discuss how we utilize them to get new results of tensor ranks and advance the current understanding of matrix multiplication complexity. This is joint work with J.M Lansberg, Austin Conner, Mateusz Michalek and Emanuele Ventura.

Feb. 15, 2022

Speaker

Cinzia Casagrande (Torino)

Title

Fano manifolds with Lefschetz defect 3

Abstract

We will talk about a structure result for some (smooth, complex) Fano varieties X, which depends on the Lefschetz defect delta(X), an invariant of X defined as follows. Consider a prime divisor D in X and the restriction r:H^2(X,R)->H^2(D,R). Then delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. If delta(X)>3, then X is isomorphic to a product SxT, where S is a surface. When delta(X)=3, X does not need to be a product, but we will see that it still has a very rigid and explicit structure. More precisely, there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. This structure theorem allows to complete the classification of Fano 4-folds with Lefschetz defect at least 3. This is a joint work with Eleonora Romano and Saverio Secci.

Feb. 22, 2022

Speaker

Irena Peeva (Cornell)

Title

The Regularity Conjecture

Abstract

Regularity is a numerical invariant that measures the complexity of the structure of homogeneous ideals in a polynomial ring. Papers of Bayer-Mumford and others give examples of families of ideals attaining doubly exponential regularity. In contrast, Bertram-Ein-Lazarsfeld, Chardin-Ulrich, and Mumford have proven that there are nice bounds on the regularity of the ideals of smooth projective varieties. As discussed in an influential paper by Bayer and Mumford (1993), the biggest missing link between the general case and the smooth case is to obtain a decent bound on the regularity of all prime ideals--the ideals that define irreducible projective varieties. The longstanding Eisenbud-Goto Regularity Conjecture (1984) predicts an elegant linear bound in terms of the degree of the variety (also called multiplicity). The conjecture was proven for curves by Gruson-Lazarsfeld-Peskine, for smooth surfaces by Lazarsfeld and Pinkham, for most smooth 3-folds by Ran, and in many other special cases. McCullough and I introduced two new techniques and used them to provide many counterexamples to the Eisenbud-Goto Regularity Conjecture. In fact, we show that the regularity of prime ideals is not bounded by any polynomial function of the degree. Starting from an arbitrary homogeneous ideal N, our ideas make it possible to construct a prime ideal whose regularity, degree, (and other numerical invariants) are expressed in terms of numerical invariants of N. The talk will discuss the concept of regularity and provide an overview of the current state of results on regularity of prime ideals.

Mar. 1, 2022

Speaker

Luís Duarte (Genoa)

Title

Ideals in a local ring under small perturbations

Abstract

Let I be an ideal of a Noetherian local ring R. We study how properties of I change for small perturbations, that is, for ideals J that are the same as I modulo a large power of the maximal ideal. In particular, assuming that J has the same Hilbert function as I, we show that the Betti numbers of J coincide with those of I. We also compare the local cohomology modules of R/J with those of R/I.

Mar. 16, 2022

Speaker

Yairon Cid-Ruiz (Ghent University)

Title

Multidegrees at the Crossroads of Algebra, Geometry, and Combinatorics

Abstract

The concept of multidegrees provides the right generalization of degree to a multiprojective setting, and its study goes back to seminal work by van der Waerden in 1929. We will give a basic introduction to the notion of multidegrees of a multiprojective variety. Then, a complete characterization for the positivity of multidegrees will be presented. Finally, we will review some important applications that follow from the use of multidegrees.

Mar. 17, 2022

Speaker

Wern Yeong (Notre Dame)

Title

Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

Abstract

A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we improve the known result that very general hypersurfaces in Pn of degree at least 2n − 2 are algebraically hyperbolic when n is at least 6 to when n is at least 5, leaving n = 4 as the only open case.

Mar. 22, 2022

Speaker

Kangjin Han (DGIST and Berkeley)

Title

Rank 3 quadratic generation of Veronese varieties

Abstract

Let $X$ be any nondegenerate projective variety over an algebraically closed field $K$ with $char(K)\neq 2$ and $L$ be any very ample line bundle on $X$. We say that $(X,L)$ satisfies property $QR(k)$ if the homogeneous ideal of $X\subset\mathbb{P}H^0(X,L)$ can be generated by quadrics of rank at most $k$. It is well-known that many classical varieties such as any Segre-Veronese embeddings, rational normal scrolls and curves of high degree satisfy $QR(4)$. In this talk, we consider rank 3 quadratic generation for Veronese varieties. We first introduce some methods to produce rank 3 quadrics and explain how to prove any Veronese variety $\nu_d(\mathbb{P}^n)$ satisfies $QR(3)$ by applying these methods. We also show some generalization of it to any variety under some positive embedding. Some families of examples with $QR(k)$ for low k will be presented in the talk as well.

Mar. 29, 2022

Speaker

Eugen Rogozinnikov (Strasbourg)

Title

Hermitian Lie groups as symplectic groups over noncommutative algebras

Abstract

In my talk, I introduce the symplectic group $\Sp_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$ and show that many classical Lie groups can be seen in this way. Of particular interest will be the classical Hermitian Lie groups of tube type and their complexifications. For these groups, I construct different models of the symmetric space in terms of the group $\Sp_2(A,\sigma)$. We obtain generalizations of several models of the hyperbolic plane and the three-dimensional hyperbolic space. This is a joint work with D. Alessandrini, A. Berenstein, V. Retakh and A. Wienhard.

Apr. 5, 2022

Speaker

Eric Jovinelly (Notre Dame)

Title

Extreme Divisors on M_{0,7} and Differences over Characteristic 2

Abstract

The cone of effective divisors controls the rational maps from a variety. We study this important object for M_{0,n}, the moduli space of stable rational curves with n markings. Fulton once conjectured the effective cones for each n would follow a certain combinatorial pattern. However, this pattern holds true only for n < 6. Despite many subsequent attempts to describe the effective cones for all n, we still lack even a conjectural description. We study the simplest open case, n=7, and identify the first known difference between characteristic 0 and characteristic p. Although a full description of the effective cone for n=7 remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions.

Apr. 12, 2022

Speaker

Wenliang Zhang (UIC)

Title

Matlis duals of local cohomology modules

Abstract

The study of Matlis duals of local cohomology modules has gained momentum in recent years, due to the work of Hellus, Hartshorne-Polini, Lyubeznik-Yildirim, etc. In this talk, I will discuss some recent results concerning the Lyubeznik-Yildirim conjecture, which asserts: Let R be a regular local ring and I be an ideal. If H^j_I(R) is non-zero, then the support of its Matlis dual is Spec(R).

Apr. 19, 2022

Speaker

Andrei Jorza (Notre Dame)

Title

What can you tell about a modular form from its second Fourier coefficient?

Abstract

The Fourier coefficients of modular forms which are eigenvectors of Hecke operators encode an enormous amount of arithmetic information. In general automorphic setting, Hecke eigenvalues determine the automorphic representation uniquely, even if you lose a positive proportion of the eigenvalues. In the case of classical modular forms of level 1, it is conjectured that just the coefficient $a_2$ suffices to define the modular form uniquely. I will present two papers with Liubomir Chiriac about the coefficients $a_2$, which bridge the world of 2-adic modular forms and that of linear approximations of logarithms.

Apr. 26, 2022

Speaker

Matthew Dyer (Notre Dame)

Title

Multiply partially representable functors

Abstract

Partially representable functors are multivariable set-valued functors which are representable in a subset of their arguments, viewed collectively as a single argument, with the complementary arguments fixed in any way. We discuss basic properties of functors which are partially representable in each of several subsets of their arguments, and structures involving them, Stricter incarnations of this notion of multiply partially representable functors include functors, (parameterized, multivariable) adjunctions and various more exotic things. Prerequisites for understanding the main ideas will be kept minimal; the talk will begin with a simple motivating example from commutative algebra, and basic notions will be introduced in the context of sets and functions.