Objects of a Phantom on a Rational Surface

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.