Date: 2022. January 21th (Fri.), Online
Time table
Our workshop will start at 9:30am. A schedule is following:
09:30-10:30 Youngrak Kim (Pusan Nat'l Univ.)
10:30-10:50 Coffee break
10:50-11:50 Kyeong-Dong Park (KIAS)
11:50-12:10 Free discussion
12:10-13:10 Jinhyung Park (Sogang Univ.)
13:10-13:30 Discussion and Wrap-up
Titles and Abstracts
1. Youngrak Kim (PNU)
Title: Ulrich bundles on cubic fourfolds
Abstract: Ulrich bundles are geometric objects corresponding to maximally generated maximal Cohen-Macaulay modules, whose existence has several interesting applications in commutative algebra, homological algebra, and more. After the pioneering works of Beauville and Eisenbud-Schreyer, existence and classification of Ulrich bundles become important questions also in projective geometry. For instance, they could help to understand the cone of cohomology tables of coherent sheaves on the underlying projective variety, determinantal representations of hypersurfaces, and determinantal representations of Cayley-Chow forms. In this talk, I will discuss construction of Ulrich bundles on smooth cubic fourfolds. Unlike smooth cubic surfaces or threefolds, the smallest possible rank of Ulrich bundles on a smooth cubic fourfold may vary when it is special, i.e., X contains certain surfaces which are not homologous to complete intersections. On the other hand, a (very) general cubic fourfold does not have an Ulrich bundle of rank < 6. I will explain how to construct a rank 6 Ulrich bundle on an arbitrary smooth cubic fourfold. This is a joint work with Daniele Faenzi.
2. Kyeong-Dong Park (KIAS)
Title: Rigidity and K-stability of Fano symmetric varieties
Abstract: Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. By the Luna-Vust theory, they are classified by involutions of semisimple algebraic groups and combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. As a result, we know that there are only six nonhomogeneous smooth projective symmetric varieties of Picard number one. In this talk, I will present results on the rigidity under Kähler deformations of smooth projective symmetric varieties of Picard number one and K-stability of some Fano symmetric vatirties; and discuss the remaining problems.
3. Jinhyung Park (Sogang U.)
Title: Asymptotic vanishing of syzygies of algebraic varieties
Abstract: About ten years ago, Ein-Lazarsfeld showed the asymptotic nonvanishing theorem of syzygies of algebraic varieties. Their result roughly says that almost all "asymptotic syzygies" of algebraic varieties are nonvanishing, and they conjectured that the remaining "asymptotic syzygies" are vanishing. This suggests that the minimal free resolutions of the graded section rings of line bundles on projective varieties have a uniform asymptotic shape as the positivity of the line bundles grows. In this talk, we confirm Ein-Lazarsfeld's conjecture on vanishing of asymptotic syzygies of algebraic varieties. We first prove the conjecture for the case of products of projective spaces, and then following Raicu's argument, we deduce the general case from this special case.
About this event...
This workshop is organized by Kangjin Han (DGIST) and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (grant no. 2021R1F1A104818611).