Algebra Seminar at UJ

Summer semester 2021

The Algebra Seminar at UJ will be taking place on Tuesday mornings at 10am (Europe/Warsaw), unless otherwise stated. This term the seminar is online over Zoom.

Abstracts

Sara Angela Filippini - Unprojections and Tom and Jerry

In the quest of finding a structure theorem for Gorenstein ideals of codimension 4 Kustin and Miller introduced ?unprojections?, which allow to construct Gorenstein rings of bigger codimension from simpler, lower codimension ones. Later Altinok and Reid recovered this procedure by studying Gorenstein rings arising from K3 surfaces and threefolds. I will discuss two families of codimension 4 Gorenstein rings, called Tom and Jerry, arising as unprojections. If time permits, I will explain how an example of T&J appears in the context of mirror symmetry for Fano varieties. This talk is based on papers by M. Reid and collaborators and A. Petracci.

Tymoteusz Chmiel - Bi-graded Koszul modules and Green's conjecture

Green's conjecture concerns syzygies of canonically embeded curves. It was proven for generic curves in characteristic 0 by Voisin at the beginning of 21st century. A different proof was discovered by Aprodu, Farkas, Papadima, Raicu and Weyman. It uses the theory of Koszul modules and has the advantage of working also in certain positive characteristics. Recently, Raicu and Sam extended this methods to the case of bi-graded modules and obtained even stronger results in this direction. In my talk I will introduce the Green's conjecture, recall the theory of Koszul modules and bi-graded Koszul modules, as well as sketch how to use it to obtain the generic Green's conjecture. The talk will be based on a paper ?Bi-graded Koszul modules, K3 carpets and Green's conjecture? by Claudiu Raicu and Steven V Sam.

Selvi Kara - Multi-Rees Algebras of Strongly Stable Ideals

In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals, the number of Borel generators of each ideal, and the degrees of Borel generators. In addition, we utilize combinatorial objects and tools such a s fiber graphs and Noetherian reduction relations to detect Koszulness of these algebras.

Alessio Sammartano - On the Hilbert scheme of points in ℙ³

In this work we study the tangent space to the Hilbert scheme Hilb^d ℙ³, motivated by conjectures of Briancon-Iarrobino stated in 1978 and by Haiman's work on Hilb^d ℙ². This is a joint work with Ritvik Ramkumar.

Anand Deopurkar - Some questions about Bridgeland stability conditions on triangulated categories discussed using the ideas from Teichmuller geometry

Some questions about Bridgeland stability conditions on triangulated categories discussed using the ideas from Teichmuller geometry.

Claudiu Raicu - De Rham (co)homology of determinantal varieties

The foundations of an algebraic De Rham (co)homology theory for possibly singular varieties in characteristic zero go back to a famous paper of Hartshorne from the 70s. When X is a complex algebraic variety, its De Rham cohomology agrees with the singular cohomology of the underlying analytic space X^an, while the De Rham homology of X computes the Borel-Moore homology of X^an. In my talk I will explain how to compute these invariants when X is a determinantal variety (generic, symmetric, or skew-symmetric), based on the study of the De Rham cohomology of local cohomology groups, and a spectral sequence featured in recent work of Hartshorne and Polini. Joint work with Andras Lorincz.