Arithmetic Geometry and Number Theory RTG Seminar: Semiorthogonal decompositions for projective plane

Seminar | November 25 | 3:10-5 p.m. | 740 Evans Hall

 Dmitrii Pirozhkov, Columbia

 Department of Mathematics

A semiorthogonal decomposition is a way to decompose a derived category into smaller components. We know many examples, but we do not really understand the constraints on the structure of an arbitrary decomposition. In this talk I will show that all semiorthogonal decompositions of the derived category of coherent sheaves on the projective plane $P^2$ arise from full exceptional collections, i.e., the known examples exhaust all possibilities. This implies that there are no phantom subcategories in $P^2$, making it the second nontrivial geometric example where we can prove this, the first being $P^1$.

In the pre-talk I will explain what a semiorthogonal decomposition of a triangulated category is, and indicate some applications of this concept. Some particularly nice decompositions come from exceptional objects and exceptional collections. I will give examples of exceptional collections. If time permits, I will also touch upon the classification result for exceptional collections on $P^2$ by Gorodentsev and Rudakov.