Algebraic Geometry/Commutative Algebra Seminar, 2019–2020

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

Sep. 4, 2019

Speaker

Mike Perlman (Notre Dame)

Title

Hodge ideals for determinantal hypersurfaces

Abstract

Hodge ideals are invariants of hypersurface singularities that arise from birational geometry and M. Saito's theory of mixed Hodge modules. We compute explicitly these ideals for hypersurfaces defined by the generic determinant. To do this we take advantage of the rich symmetry of their log resolutions, the spaces of complete collineations. This is part of joint work in progress with Mircea Mustata and Claudiu Raicu.

Sep. 12, 2019

Speaker

Chris Eur (U.C. Berkeley)

Title

Simplicial generation of Chow rings of matroids

Abstract

Matroids generalize combinatorially the notion of linear independence and graphs, and has recently seen a breakthrough via the development of Hodge theory on Chow rings of matroids by Adiprasito, Huh, and Katz. We introduce a new presentation of the Chow ring of a matroid whose variables now admit a combinatorial interpretation via the theory of matroid quotients and display a geometric behavior analogous to that of nef classes on smooth projective varieties. We discuss various applications, including the recovery of the Hodge theory of matroids. This is joint work with Spencer Backman and Connor Simpson.

Sep. 18, 2019

Speaker

Eric Riedl (Notre Dame)

Title

Linear subvarieties of hypersurfaces and unirationality

Abstract

The de Jong-Debarre Conjecture predicts that the space of lines on any smooth hypersurface of degree d <= n in P^n has dimension 2n-d-3. We prove this conjecture for n > 2d, improving on the previously-known exponential bounds. We prove an analogous result for k-planes, and use this generalization to prove that an arbitrary smooth hypersurface is unirational if n > 2^{d!}. This is joint work with Roya Beheshti.

Sep. 25, 2019

Speaker

Remi Jaoui (Notre Dame)

Title

On the solutions of very general planar algebraic vector fields

Abstract

It is a theorem from Landis and Petrovskii from the fifties that the only algebraic solutions of a very general planar algebraic vector field are the stationary solutions. The non-stationary solutions of a very general vector field are therefore given by transcendental functions but few things are known about the nature of these transcendental functions. For example, can they be expressed using only “classical“ transcendental functions such as exponentials, logarithms, Weierstrass's elliptic functions? In my talk, I will describe a stronger non-integrability result (irreducibility in the sense of Painlevé-Umemura) for very general planar algebraic vector fields of degree greater or equal to three, based on a combination of methods from model-theory and from the local theory of vector fields singularities.

Oct. 2, 2019

Speaker

Kostya Timchenko (Notre Dame)

Title

Characteristic cycles for K-orbits on Grassmannians

Abstract

Bressler-Finkelberg-Lunts showed that the characteristic cycle of the IC-sheaf associated to any Schubert variety in the Grassmannian is irreducible. We consider the case of K-orbits on Grassmannians, where K is the connected component of the fixed point set of an involution of GL(n). We compute the characteristic cycles associated to these orbits. They are irreducible for K = GL(p)xGL(q) and Sp(2n). When K is SO(n), roughly half of them are irreducible.

Oct. 9, 2019

Speaker

Anand Patel (Oklahoma State)

Title

Projection and Ramification

Abstract

When a projective variety is linearly projected onto a projective space of the same dimension, a ramification divisor appears. In joint work with Anand Deopurkar and Eduard Duryev, we study very basic questions about the map which sends a projection to its ramification divisor. I will present proven results, more open problems, and if time permits, some intriguing numerology.

Oct. 16, 2019

Speaker

Sayanta Mandal (UIC)

Title

Restricted tangent bundle of Grassmannian to rational curves

Abstract

Let n≥4, 2≤r≤n−2 and e≥1. We show that the intersection of the locus of degree e morphisms from P^1 to G(r,n) with the restricted universal sub-bundles having a given splitting type and the locus of degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree e morphisms from P^1 to G(r,n) with the restricted tangent bundle having a given splitting type can have arbitrarily large number of irreducible components.

Oct. 17, 2019

Speaker

Gavril Farkas (Humboldt University)

Title

The global geometry of the moduli space of curves

Abstract

The moduli space of curves M_g is the universal parameter space for Riemann surfaces of given genus. Its study has been initiated by Riemann in 1857 and it has been a long-standing problem to describe the nature of the moduli space as an algebraic variety. I will survey the history of the problem starting with Severi's conjecture from 1915 predicting that M_g is always unirational, continuing with the work of Harris and Mumford spectacularly disproving Severi's conjecture and finally discussing very recent results obtained jointly with Jensen and Payne which settle this problem in two of the most interesting remaining cases, those of genus 22 and 23.

Oct. 30, 2019

Speaker

Zachary Flores (Colorado State)

Title

The Weak Lefschetz Property for Cohomology Modules of Vector Bundles on P^2

Abstract

The study of Lefschetz properties is about understanding how linear forms act on finite length modules. However, results on Lefschetz properties for even broad classes of Artinian algebras remain elusive, but, through a slightly different perspective, we discuss a useful generalization of an important result that showed complete intersections in characteristic zero and codimension three have the Weak Lefschetz Property.

Nov. 6, 2019

Speaker

Juliette Bruce (University of Wisconsin-Madison)

Title

Semi-Ample Asymptotic Syzygies

Abstract

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

Nov. 14, 2019

Speaker

Antoni Rangachev (University of Chicago)

Title

Generalized Smoothability

Abstract

In this talk I will introduce a class of singularities that generalizes the class of smoothable singularities: these are all singularities that admit deformations to deficient conormal (dc) singularities. I will discuss how this new class arises from problems in differential equisingularity and how it relates to the vanishing of the local volume of a line bundle. Using Thom's transversality, Whitney stratifications and Lagrangian geometry I will show that all smoothable, codimension 2 Cohen-Macaulay, codimension 3 Gorenstein, almost complete intersections, and more generally determinantal and Pfaffian singularities admit deformations to dc singularities.

Nov. 20, 2019

Speaker

Alessio Sammartano (Notre Dame)

Title

On the tangent space to the Hilbert scheme of points in P^3

Abstract

In this work we study the tangent space to the Hilbert scheme Hilb^d(P^3), which parametrizes 0-dimensional subschemes of P^3 of degree d. We are mainly motivated by conjectures of Briancon-Iarrobino on the most singular point in the Hilbert scheme, and by Haiman’s work on the Hilbert scheme of P^2. This is a joint work with Ritvik Ramkumar.

Nov. 22, 2019

Speaker

Laurențiu Maxim (University of Wisconsin-Madison)

Title

Topological methods in applied algebra and algebraic statistics

Abstract

Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, engineering, and statistics. One way to solve this problem is to determine the critical points of an objective (e.g., distance) function on the model. In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of polynomial equations. The number of critical points of the squared Euclidean distance function on the (complexification of an) algebraic model is a measure of the algebraic complexity for the nearest point problem, called the Euclidean distance degree. In this talk, I will present some models from linear algebra, computer vision and statistics that may be described as algebraic sets, and I will discuss a new topological method for determining the Euclidean distance degree. As applications, I will discuss the solution to an open problem in computer vision of determining the Euclidean distance degree of the multiview variety, and I will answer positively a conjecture of Aluffi-Harris concerning the Euclidean distance degree of projective varieties. Such projective models appear naturally in low rank matrix approximation, formation shape control and all across algebraic statistics.

Dec. 4, 2019

Speaker

John Sheridan (Stony Brook)

Title

Continuous families of divisors on symmetric powers of curves

Abstract

For X a smooth projective variety, we consider its set of effective divisors in a fixed cohomology class. This set naturally forms a projective scheme and if X is a curve, this scheme is a smooth, irreducible variety (fibered in linear systems over the Picard variety). However, when X is of higher dimension, this scheme can be singular and reducible. We study its structure explicitly when X is a symmetric power of a curve.

Jan. 29, 2020

Speaker

Juan Migliore (Notre Dame)

Title

The singular locus of a hyperplane arrangement in P^3, and liaison

Abstract

This talk will describe joint with with Uwe Nagel and Hal Schenck. Let A be a hyperplane arrangement in P^3. (Much of the work actually holds in P^n.) Let J be the Jacobian ideal of A. This ideal is usually not unmixed, or even saturated. We consider two associated ideals, J^{top} and \sqrt{J}, both of which ARE unmixed and define equidimensional curves X^{top} and X^{red} in P^3. These curves both have a claim to being called the (unmixed) singular locus of A. If a certain fairly mild combinatorial property (*) of the incidence lattice of A is satisfied, both X^{top} and X^{red} are arithmetically Cohen-Macaulay (ACM). If (*) does not hold, either or both of these curves may fail to be ACM. In fact, they can be made to fail to be ACM by as much as you like (as measured by the dimension of the Hartshorne-Rao module). The proofs use tricks from liaison, which in turn prompt liaison-related questions. We will briefly review facts from liaison along the way.

Feb. 5, 2020

Speaker

Erika Ordog (Duke)

Title

Minimal resolutions of monomial ideals

Abstract

The problem of finding minimal free resolutions of monomial ideals in polynomial rings has been central to commutative algebra ever since Kaplansky raised the problem in the 1960s and his student, Diana Taylor, produced the first general construction in 1966. The ultimate goal along these lines is a construction of free resolutions that is universal -- that is, valid for arbitrary monomial ideals -- canonical, combinatorial, and minimal. This talk describes a solution to the problem valid in charactersitic 0 and almost all positive charactersitics.

Feb. 12, 2020

Speaker

Claudia Polini (Notre Dame)

Title

Blowup algebras of ideals of minors of sparse matrices

Abstract

The study of rings and more generally of varieties that are defined by determinantal ideals of generic matrices has been a central topic of commutative algebra and algebraic geometry. We consider the ideal I generated by maximal minors of sparse matrices of size 2 by n. Using the theory of Sagbi bases we describe the defining equations of the Rees algebra of such ideals. In particular we show that I is of fiber type and the special fiber ring is defined by the Plucker relations. We obtain many consequences: the Rees algebra of I has rational singularities if the field has characteristic zero and is F-rational if the field is of positive characteristic. In particular, the Rees algebra is a Cohen-Macaulay, normal domain. In addition, the Plucker relations together with the linear relations form a Groebner basis for the defining ideal of the Rees ring, hence the latter is Koszul and I has linear powers. This is joint work with Ela Celikbas, Emilie Dufresne, Louiza Fouli, Elisa Gorla, Kuei-Nuan Lin, and Irena Swanson.

Feb. 27, 2020

Speaker

Michael Brown (Wisconsin)

Title

Standard Conjecture D for matrix factorizations

Abstract

In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. They have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general statements, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations.

Mar. 4, 2020

Speaker

Wenliang Zhang (UIC)

Title

Asymptotic vanishing theorems for lci projective schemes

Abstract

I will discuss some (asymptotic) vanishing theorems for projective schemes that are locally complete intersection; some are only valid in characteristic 0 while others hold over fields of arbitrary characteristics. This is based on joint work with Bhatt, Blickle, Lyubeznik, and Singh.