2026 Theses Doctoral

Objects of a Phantom on a Rational Surface

Mattoo, Amal

Johannes Krah showed that the blowup of 𝚸² 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 𝚸², and we show that there exists a co-connective dg-algebra whose derived category is a phantom. We also extend these results to similarly constructed phantoms on other rational surfaces, and we show how our tools can be used to compute Hochschild cohomology including its product structure.