Admissible subcategories of del Pezzo surfaces

Admissible subcategories are building blocks of semiorthogonal decompositions. Many examples of them are known, but few general properties have been proved, even for admissible subcategories in the derived categories of coherent sheaves on basic varieties such as projective spaces. We use a relation between admissible subcategories and anticanonical divisors to study admissible subcategories of del Pezzo surfaces. We show that any admissible subcategory of the projective plane has a full exceptional collection, and since all exceptional objects and collections for the projective plane are known, this provides a classification result for admissible subcategories. We also show that del Pezzo surfaces of degree at least three do not contain so-called phantom subcategories. These are the first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms.