Graduate Algebraic Geometry Seminar Fall 2022

About

This is a graduate student seminar at UIC in algebraic geometry, where we share exciting mathematics every week. We meet every Monday 4:00-5:00 pm at SEO 427.

Week 12: Nov 7, 2022

Sixuan Lou: Examples of rank 2 vector bundles on P^3 with c_1 = 0


We will compute a few examples of moduli spaces of rank 2 vector bundles on P^3 with vanishing c_1. We will describe some general strategies of constructing components in the moduli space.


References:

Okonek, C.; Schneider, M.; Spindler, H. Vector Bundles on Complex Projective Spaces; Springer Basel: Basel, 1980; ISBN 978-3-0348-0150-8.

Ein, L. Generalized Null Correlation Bundles. Nagoya Mathematical Journal 1988, 111, 13–24, doi:10.1017/S0027763000000970.

Almeida, C.; Jardim, M.; Tikhomirov, A.; Tikhomirov, S. New Moduli Components of Rank 2 Bundles on Projective Space. Sb. Math. 2021, 212, 1503–1552, doi:10.1070/SM9490.

Week 11: Oct 31, 2022

Ben Gould: Stable rank 2 bundles and curves on P^3


We introduce the results of Hartshorne's paper 'Stable bundles of rank 2 on P3', which describes a precise relationship between slope-stable vector bundles of rank 2 on P3 and curves in P3. This is the first of a series of talks on the topic of rank 2 vector bundles on projective spaces.


References:

Hartshorne, R. Stable Vector Bundles of Rank 2 on P^3. 52.

Week 10: Oct 24, 2022

Shravan Patankar: Vanishing of Tors of absolute integral closures in equicharacteristic zero

Let R be a N-graded ring of dimension 2 finitely generated over a field k of characteristic 0. I show that if Tor_i(R^+, k) = 0 then R is regular. This answers a question of Bhatt, Iyangar, and Ma. The answer, astonishingly, uses geometric techniques and a simple application of "almost mathematics".

Week 8: Oct 17, 2022

Yeqin Liu: Cohomology of spherical vector bundles on K3 surfaces

On K3 surfaces, spherical vector bundles are the ones without deformations. In this talk I introduce a way to compute their exact numbers of global sections by using Bridgeland stability. I will show many examples.

Week 5: Sept 26, 2022

Greg Taylor: Syzygies of Curves and Beyond

In this talk, we give an overview of some classic results on the equations defining projective varieties (emphasizing the curve case). Then we shift our focus to current research topics and open questions.

Week 4: Sept 19, 2022

Sixuan Lou: Hodge theory and derived categories of cubic fourfolds

Based on Ben and Junyan's talks, I will explain the work of Addington and Thomas that establishes the link between the Hodge theory and the derived category of cubic fourfolds.

References:

  • N. Addington, R. Thomas - Hodge theory and derived categories of cubic fourfolds

  • D. Huybrechts - The geometry of cubic hypersurfaces (draft)

Week 3: Sept 12, 2022

Junyan Zhao: Special Cubic 4-folds, K3 surfaces and Rationality

Rationality problems have been attracting people for several decades. One of the most unknown non-trivial examples is cubic fourfolds. Unfortunately, cubic fourfoulds vary in their behavior. In this talk, I will give a brief introduction of those families which are proven to be rational. When working under this framework, two ingredients are essentially useful: K3 surfaces and Hodge theory. We will end up with the most famous conjecture of Kusnetsov on the rationality of cubic fourfolds.

References:

  • Arnaud Beauville, BrendanHassett, AlexanderKuznetsov, AlessandroVerra - Rationality Problems in Algebraic Geometry

  • Brendan Hassett - Special Cubic Fourfolds

Week 2: Sept 7, 2022

Ben Gould: An introduction to Kuznetsov components of derived categories of Fano varieties

In dimension two, Bridgeland stability in derived categories has produced very effective tools for doing algebraic geometry. We saw a very good example in Yeqin's talk about the Feyzbakhsh-Mukai program last week. In higher dimensions there are also Bridgeland stability conditions, but they are much more difficult to use, and many fewer applications are known.

When dealing with Fano varieties of dimension three or four, we can get around this problem by examining Kuznetsov components of the derived category. Roughly speaking, we 'subtract' the data of a small number of exceptional bundles (usually line bundles), and study the remaining category. This category 'looks like' the derived category of a curve or surface, and we can use our old techniques to study it and draw consequences for the original Fano. This smaller category is the Kuznetsov component, which we will discuss in this talk.

References:

  • Bayer, Lahoz, Macri, Stellari - Stability conditions on Kuznetsov components

  • Kuznetsov - Derived categories of quadric fibrations and intersections of quadrics

  • Bernadara, Macri, Mehrota, Stellari - A categorical invariant of cubic threefolds (start here)

Week 1: Aug 31, 2022

Yeqin Liu: Mukai's program

Mukai's program reconstructs K3 surfaces from some curves on them. The statement itself is classical, however the proof involves Bridgeland stability in a non-trivial way.

(This is another phenomenon of reconstruction from Brill-Noether locus, which fits into the last topic of last gags.)

References:

For the Mukai's program:

  • S. Feyzbakhsh. Mukai’s program (reconstructing a k3 surface from a curve) via wall-crossing

For Bridgeland stability:

  • T. Bridgeland. Stability conditions on triangulated categories

  • T. Bridgeland. Stability conditions on k3 surfaces

  • A. Bayer and E. Macrı̀. Mmp for moduli spaces of sheaves on k3s via wall-crossing: nef and movable cones, lagrangian fibrations

  • A. Bayer and E. Macrı̀. Projectivity and birational geometry of bridgeland moduli spaces