Objects of a Phantom on a Rational Surface

TL;DR

This paper explores the rich geometric and algebraic structures within a phantom subcategory of the derived category of the blowup of P^2 at 10 points, revealing new objects and deformation properties.

Contribution

It constructs explicit objects in the phantom, analyzes their deformation theory, and links the phantom to a co-connective dg-algebra, advancing understanding of phantom categories.

Findings

Constructed a strong generator in the phantom

Identified projections of skyscraper sheaves within the phantom

Discovered a family of objects with two nonzero cohomology sheaves

Abstract

Johannes Krah showed that the blowup of $\mathbf{P}^{2}$ in $10$ general points admits a phantom subcategory. We construct three types of objects in such a phantom: a strong generator, projections of skyscraper sheaves, and a family of objects with two nonzero cohomology sheaves. We study the deformation theory of these objects to show that the phantom contains rich geometry, such as encoding the blowdown map to $\mathbf{P}^{2}$. We also show that there exists a co-connective dg-algebra whose derived category is a phantom.