All Quiet on the Exceptional Locus

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.