Admissible subcategories of del Pezzo surfaces

TL;DR

This paper classifies admissible subcategories of derived categories on del Pezzo surfaces, showing their structure and absence of phantom subcategories, with implications for understanding derived categories of higher-dimensional varieties.

Contribution

It provides a complete classification of admissible subcategories on the projective plane and proves the nonexistence of phantom subcategories on del Pezzo surfaces.

Findings

Classified all admissible subcategories of the projective plane.

Proved that del Pezzo surfaces do not contain phantom subcategories.

Showed admissible subcategories supported on (-1)-curves are generated by sheaves.

Abstract

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (-1)-curve in a surface is generated by some twist of the structure sheaf of that curve.