Exceptional collections on nonminimal Enriques surfaces

Abstract:

By Orlov’s formula, the derived category of blow up $X = \operatorname {Bl}_pX’ \to X’$ contains $\operatorname {D}^{\mathsf {b}}(X’)$ as a semiorthogonal component. This raises an interesting question: does there exist a variety $X’$ such that $\operatorname {D}^{\mathsf {b}}(X’)$ does not admit an exceptional collection of maximal length, but $\operatorname {D}^{\mathsf {b}}(X)$ admits an exceptional collection of maximal length? We give such an example when $X’$ is a minimal Enriques surface.