The Hirzebruch isomorphism for exotic noncommutative surfaces

Abstract: The isomorphism between the first Hirzebruch surface and the blowup of the projective plane in a point is a well-known result, due to Hirzebruch. From a numerical classification of Grothendieck groups of rank 4 which behave like the Grothendieck group of a smooth projective surface we expect the existence of exotic noncommutative surfaces, which are surfaces not obtained via deforming commutative surfaces. There exist two constructions: one as an asymmetric noncommutative P^1-bundle (due to de Thanhoffer--Presotto), one as a fat point blowup (joint work with Presotto). I will explain how a Hirzebruch isomorphism for these two families of surfaces exists as an equivalence of (derived) categories, and how this is related to some very classical geometry of linear systems. This is joint work with Dennis Presotto and Michel Van den Bergh.

algebraic geometry

Audience: researchers in the topic

Derived seminar

Series comments: https://ed-ac-uk.zoom.us/j/89993982042

Password: a simply-connected two-dimensional variety with trivial canonical bundle (omit the space)