Autoequivalences of Blow-Ups of Minimal Surfaces

Abstract

Let X be the blow-up of the projective plane in a finite set of very general points. We deduce from the work of Uehara that X has only standard autoequivalences, no nontrivial Fourier-Mukai partners, and admits no spherical objects. If X is the blow-up of the projective plane in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if X is a blow-up of finitely many points in a minimal surface of nonnegative Kodaira dimension which contains no (-2)-curves. Independently, we characterize spherical objects on blow-ups of minimal surfaces of positive Kodaira dimension.