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
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COVID19 Update
As a result of Covid19, the seminar for this semester will be a mix of virtual and inperson 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 FourierMukai 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://wwwusers.cse.umn.edu/~cberkesc/CA/CA2021.html]  
November 5


November 12  TALK AT NONSTANDARD TIME  Jinhyung Park at 9:00am (Zoom)  Asymptotic vanishing of syzygies of algebraic varieties  
November 12  Daniel Erman at usual time (2:30pm)  
November 19  Ritvik Ramkumar (UC Berkeley; Zoom)  Rational singularities of nested Hilbert schemes.  Daniel  
November 26  Thanksgiving  
December 3  Eric Ramos  Equivariant logconcavity  
December 10  
April 8  Haydee Lindo  Daniel 
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 AndreottiMayer, HarrisArbarello and Green, which tell us that canonical curves are defined by quadrics of rank four.
Michael Brown
Title: Tate resolutions as noncommutative FourierMukai transforms
Abstract: This is joint work with Daniel Erman. The classical BernsteinGel'fandGel'fand (or BGG) correspondence gives an equivalence between the derived categories of a polynomial ring and an exterior algebra. It was shown by EisenbudFløystadSchreyer 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 FourierMukai 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 LazarsfeldMukai 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.
Jinhyung Park
Title: Asymptotic vanishing of syzygies of algebraic varieties
Abstract: In this talk, we show EinLazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with EinLazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows.
Daniel Erman
Title: The geometry of virtual syzygies
Abstract: One of the foundational results connecting syzygies with algebraic geometry properties was Mark Green’s result on N_p conditions for smooth curves of high degree. A modern and streamlined proof of this result comes via Green’s Linear Syzygy Theorem. I will discuss very recent work with Michael Brown which proves a Multigraded Linear Syzygy Theorem and uses this to obtain the first known examples of “virtual" N_p conditions for smooth curves of high degree in other toric varieties. This is joint work with Michael Brown.
Ritvik Ramkumar
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 0dimensional 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 squarefree Gröbner degenerations as well as the KempfWeyman geometric technique. This is joint work with Alessio Sammartano.