# Algebra and Algebraic Geometry Seminar Fall 2021

The Seminar will take place on Fridays at 2:30 pm, either virtually (via Zoom) or in person, in room B235 Van Vleck.

## Contents

## Algebra and Algebraic Geometry Mailing List

- Please join the AGS mailing list by sending an email to ags+join@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## COVID-19 Update

As a result of Covid-19, the seminar for this semester will be a mix of virtual and in-person talks. The default Zoom link for the seminar is https://uwmadison.zoom.us/j/9502605167 (sometimes we will have to use a different meeting link, if Michael K cannot host that day).

## Fall 2021 Schedule

date | speaker | title | host/link to talk | |
---|---|---|---|---|

September 24 | Michael Kemeny (local, in person) | The Rank of Syzygies | ||

October 1 | Michael K Brown (Auburn University) | Tate resolutions as noncommutative Fourier-Mukai transforms | Daniel | |

October 8 | Yi (Peter) Wei (local) | Geometric Syzygy Conjecture in char p, with reveries from Ogus’ result on a versal deformation of K3 surfaces | Michael | |

October 15 | Michael Perlman (Minnesota; virtual) | Mixed Hodge structure on local cohomology with support in determinantal varieties | Daniel | |

October 22 | Ritvik Ramkumar (UC Berkeley) | Something about Hilbert schemes, probably | Daniel | |

October 29 | CA+ meeting [ https://www-users.cse.umn.edu/~cberkesc/CA/CA2021.html] | |||

November 5 | Eric Ramos | Equivariant log-concavity | ||

November 12 | ||||

November 19 | ||||

November 26 | Thanksgiving | |||

December 3 | ||||

December 10 |

## Abstracts

### Speaker Name

### Michael Kemeny

Title: The Rank of Syzygies

Abstract: I will explain a notion of *rank* for the relations amongst the equations of a projective variety. This notion generalizes the classical notion of rank of a quadric and is just as interesting!
I will spend most of the talk developing this notion but will also explain one result which tells us that, for a randomly chosen canonical curve, you expect all the linear syzygies to have the lowest possible
rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank *four*.

### Michael Brown

Title: Tate resolutions as noncommutative Fourier-Mukai transforms

Abstract: This is joint work with Daniel Erman. The classical Bernstein-Gel'fand-Gel'fand (or BGG) correspondence gives an equivalence between the derived categories of a polynomial ring and an exterior algebra. It was shown by Eisenbud-Fløystad-Schreyer in 2003 that the BGG correspondence admits a geometric refinement, which sends a sheaf on projective space to a complex of modules over an exterior algebra called a Tate resolution. The goal of this talk is to reinterpret Tate resolutions as noncommutative analogues of Fourier-Mukai transforms, and to discuss some applications.

### Peter Wei

Title: Geometric Syzygy Conjecture in char p, with reveries from Ogus’ result on a versal deformation of K3 surfaces

Abstract: We aim to study syzygies of canonical curves in char p. I will briefly introduce how to translate the questions on curves to questions on K3 surfaces, where the Lazarsfeld-Mukai bundle plays a great role. I will show how to use Ogus’ result on a versal deformation of K3 surfaces, to help us resolve the case for a general K3 surface. And finally, I will sketch the proof of Geometric Syzygy Conjecture for even genus curve assuming an effective lower bound on the characteristics.

### Michael Perlman

Title: Mixed Hodge structure on local cohomology with support in determinantal varieties

Abstract: Given a closed subvariety Z in a smooth complex variety, the local cohomology modules with support in Z are functorially endowed with structures as mixed Hodge modules, implying that they are equipped with Hodge and weight filtrations that subtly measure the singularities of Z. We will discuss new calculations of these filtrations in the case when Z is a generic determinantal variety. As an application, we obtain the Hodge ideals for the determinant hypersurface. Joint work with Claudiu Raicu.