All Quiet on the Exceptional Locus

TL;DR

This paper proves that admissible subcategories supported on the exceptional locus of a birational morphism of smooth projective surfaces are generated by finite exceptional collections, with implications for phantom categories.

Contribution

It establishes that such subcategories are generated by finite exceptional collections and rules out the existence of phantom categories with proper support on these surfaces.

Findings

Admissible subcategories supported on the exceptional locus are generated by finite exceptional collections.

No nonzero phantom or quasi-phantom subcategories can have proper support on these surfaces.

The proof uses a combination of splitting lemmas, Orlov's blow-up formula, and support theorems.

Abstract

We study admissible subcategories of the bounded derived category of a smooth projective surface that are supported on the exceptional locus of a birational morphism. We prove that if $f:X\to Y$ is a birational morphism of smooth projective surfaces, then every admissible subcategory of $D^b(X)$ supported on $\operatorname{Exc}(f)$ is generated by a finite exceptional collection. Moreover, if $K_Y$ is nef, then the same conclusion holds for every admissible subcategory of $D^b(X)$ supported on a proper closed subset of $X$. As a consequence, no nonzero phantom or quasi-phantom subcategory on such a surface can have proper support. The proof combines a splitting lemma for admissible subcategories inside a semiorthogonal decomposition with a single exceptional block, Orlov's blow-up formula, and Pirozhkov's support theorem.