On derived categories of nonminimal Enriques surfaces

TL;DR

This paper investigates the structure of derived categories of nonminimal Enriques surfaces, providing examples where blow-ups admit maximal-length exceptional collections despite the original surface not having one.

Contribution

It constructs explicit examples of minimal Enriques surfaces whose blow-ups have maximal-length exceptional collections, addressing a key question in derived category theory.

Findings

Constructed examples of Enriques surfaces with specific derived category properties

Showed that blow-ups can admit maximal-length exceptional collections even if the original does not

Provided insights into the semiorthogonal decompositions of nonminimal surfaces

Abstract

By Orlov's formula, the derived category of blow up must contain the original variety as a semiorthogonal component. This arises an interesting question: does there exist a variety $X$ such that $\operatorname{D}^{\sf b}(X)$ does not admit an exceptional collection of maximal length, but $\operatorname{D}^{\sf b}(\operatorname{Bl}_{x} X)$ admits such a collection? We give such an example where $X$ is a minimal Enriques surface.