Exceptional collections on the Hilzebruch surface of degree 2

Hokuto Uehara (Tokyo Metropolitan University)

24-Dec-2020, 10:00-11:00 (3 years ago)

Abstract: The purpose of my talk is to clarify the structure of exceptional collections of the derived category of coherent sheaves on the Hirzebruch surface of degree 2 (a special weak del Pezzo surface), to the extent it is understood by Orlov and Kuleshov for del Pezzo surfaces. (1) First, we prove that for any exceptional object in it, one can find an autoequivalence which sends it to an exceptional vector bundle. This result was conjectured by Shinnosuke Okawa and the speaker in 2015. (2) Refining the method of the proof of the above result, and based on a deformation argument, we prove that the braid group on 4 strands acts transitively (up to shifts) on the set of exceptional collections of length 4. This is a special case of an old conjecture by Bondal and Polishchuk. (3) We also prove that any exceptional collection can be extended to a full exceptional collection. My talk is based on a joint work with Shinnosuke Okawa (Osaka) and Akira Ishii (Nagoya).

algebraic geometry

Audience: researchers in the topic


ZAG (Zoom Algebraic Geometry) seminar

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