Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Abstract

Let X be any rational surface. We construct a tilting bundle T on X. Moreover, we can choose T in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra A. The construction starts with a full exceptional sequence of line bundles on X and uses universal extensions. If X is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups Ext q for q≥2 vanishing, then X also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.