One-day Workshop on Derived Category

Intensive Lecture

• Giancarlo Urzúa _ Pontificia Universidad Católica de Chile

Categorical aspects of the Kollár--Shepherd-Barron correspondence

1. W-surfaces and their birational geometry

2. N-resolutions and the Kollár--Shepherd-Barron correspondence

3. S.o.d.'s from N-resolutions: results and open problems

Contributed Talk

• Kyoung-Seog Lee _ University of Miami

Derived categories and VGIT

Title and Abstract

• Giancarlo Urzúa

Categorical aspects of the Kollár--Shepherd-Barron correspondence

1. W-surfaces and their birational geometry 

Brief introduction to W-surfaces and their properties. Examples from degenerations of the projective plane and Enriques surfaces. The MMP machinery (including flips and divisorial contractions) and their outcomes, Mori trains. 

2. N-resolutions and the Kollár--Shepherd-Barron correspondence

Irreducible components of the deformation space of a cyclic quotient singularity and the correspondences, which introduce N-resolutions. General set-up with Wahl resolutions. Zero continued fractions and algorithm for M-resolutions. Existence and uniqueness of N-resolution via continued fractions. The example from a minimal resolution. 

3. S.O.D.'s from N-resolutions: results and open problems

Another way to find the N-resolution of an M-resolution, and the braid group action. The set-up for the main theorems about Hacking exceptional collections. Some applications (e.g. to every simply-connected Dolgachev surface), and some open questions (e.g. admissible embeddings of derived categories of modules over quiver algebras into derived categories of surfaces). 

• Kyoung-Seog Lee

Derived categories and VGIT

In this talk, I will review the works of Halpern-Leistner and Ballard-Favero-Katzarkov, studying how derived categories change under the VGIT. Then I will discuss how one can use these works to study derived categories of certain interesting algebraic varieties. The last part of this talk is based on joint work with Han-Bom Moon.

Timetable

• 10:30–11:20: Urzúa 1

• 11:30–12:20: K. Lee (via zoom:  313 252 8553)

• 14:00–14:50: Urzúa 2

• 15:00–15:50: Urzúa 3

Organizer

• Heesang Park _ Konkuk University

• Dongsoo Shin _ Chungnam National University

Support

• Basic Research Laboratory: ASADCS

• Research Institute of Basic Sciences, Konkuk University

Contact

• Dongsoo Shin: dsshin@cnu.ac.kr