Mathematical Research Letters

Volume 28 (2021)

Number 1

Discriminants of stable rank two sheaves on some general type surfaces

Pages: 245 – 270



Benjamin Schmidt (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Benjamin Sung (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)


We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics.

B. Schmidt has been supported by an AMS-Simons Travel Grant.

B. Sung has been supported by NSF RTG Grant DMS-1645877 and the NSF Graduate Research Fellowship under grant DGE-1451070.

Received 16 January 2019

Accepted 5 June 2019

Published 24 May 2022