# 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.

## 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
Tuesday, Oct. 12 Jenna Tarasova (Purdue) TBA
Tuesday, Oct. 19 No seminar (Fall break)
Tuesday, Oct. 26 Geoffrey Smith (UIC) TBA
Tuesday, Nov. 2 Hang Huang (Texas A&M) TBA
Tuesday, Nov. 9 Brian Harbourne (Nebraska) TBA
Tuesday, Nov. 16 Jonathan Montaño (New Mexico State) TBA
Tuesday, Nov. 23 Gregory Taylor (UIC) TBA
Tuesday, Nov. 30 Bernd Ulrich (Purdue) TBA
Tuesday, Dec. 7 Bernd Ulrich (Purdue) TBA

## 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.