# 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.

Abstracts can be found below.

## 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.

Date Speaker Title
Tuesday, Jan. 18
Tuesday, Jan. 25 Boming Jia (Chicago) Geometry of the Affine Closure of T^*(SL_n/U)
Tuesday, Feb. 1
Tuesday, Feb. 8
Tuesday, Feb. 15
Tuesday, Feb. 22
Tuesday, Mar. 1
Tuesday, Mar. 8 No seminar (Spring Break)
Tuesday, Mar. 15
Tuesday, Mar. 22 Kangjin Han (DGIST and Berkeley) TBA
Tuesday, Mar. 29
Tuesday, Apr. 5
Tuesday, Apr. 12 Wenliang Zhang (UIC) TBA
Tuesday, Apr. 19
Tuesday, Apr. 26

## 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.

Date Speaker Title
Tuesday, Aug. 31 Zhao Gao (Notre Dame) Cohomology of line bundles on the incidence correspondence
Tuesday, Sep. 7 Evan O'Dorney (Notre Dame) Reflection theorems for number rings
Tuesday, Sep. 14 Keller VandeBogert (Notre Dame) Iterated Mapping Cones for Strongly Koszul Algebras
Tuesday, Sep. 21 Ritvik Ramkumar (Berkeley) Rational singularities of nested Hilbert schemes
Tuesday, Sep. 28 Matthew Weaver (Purdue) Rees Algebras of Codimension Three Gorenstein Ideals of Hypersurface Rings and their Defining Equations
Tuesday, Oct. 5 Andres Fernandez Herrero (Cornell) Moduli of sheaves via affine Grassmannians
Tuesday, Oct. 12 Jenna Tarasova (Purdue) Residual Intersections of Determinantal Ideals of $2\times n$ Matrices
Tuesday, Oct. 19 No seminar (Fall break)
Tuesday, Oct. 26 Geoffrey Smith (UIC) Very free rational curves in Fano varieties
Thursday, Oct. 28
2:30–3:30pm
Note special day/time
Kevin Tucker (UIC) Splinter rings and Global +-regularity
Tuesday, Nov. 2 Hang Huang (Texas A&M) Tensor Ranks and Matrix Multiplication Complexity
Tuesday, Nov. 9 Brian Harbourne (Nebraska) The concept of geproci subsets of projective 3-space
Tuesday, Nov. 16 Jonathan Montaño (New Mexico State) Blowup algebras of determinantal ideals in prime characteristic
Tuesday, Nov. 16
3:30–4:30pm
Note special time
Qaasim Shafi (Imperial College) Gromov-Witten Invariants of Blow-Ups
Tuesday, Nov. 23 Gregory Taylor (UIC) Asymptotic syzygies of secant varieties of curves
Tuesday, Nov. 30 Richard Birkett (Notre Dame) Dynamically Stabilising Birational Surface Maps: Two Methods
Tuesday, Dec. 7 Emanuela Marangone (Notre Dame) The non-Lefschetz locus for vector bundle of rank $2$ on $\mathbb{P}^2$

## 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
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.