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arXiv · 2604.16277

All Quiet on the Exceptional Locus

Ari Krishna
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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.

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Ari Krishna. 2026-04-17. All Quiet on the Exceptional Locus. https://arxiv.org/abs/2604.16277

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