A Looming of phantoms

TL;DR

This paper constructs new examples of phantom categories within the derived categories of specific rational surfaces, advancing understanding of their semiorthogonal decompositions and proposing conjectures for broader classes.

Contribution

It introduces novel phantom categories as semiorthogonal components in derived categories of certain rational surfaces and conjectures their existence in other cases.

Findings

Constructed phantom categories for blowups of the plane and Hirzebruch surface.

Proposed conjectures for phantom categories in other rational surfaces.

Extended the understanding of semiorthogonal decompositions in algebraic geometry.

Abstract

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of the second Hirzebruch surface at 9 points in general position. We also pose conjectures about the existence of phantom subcategories in the derived categories of other rational surfaces, obtained as the blowups of the other Hirzebruch surfaces.