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: projections of skyscraper sheaves, a strong generator, 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.
Product-quotient surfaces form an interesting class of surfaces of general type. We explore the stability manifold of these surfaces. In cases where the group action is not free, we construct non-geometric stability conditions.
I will discuss recent joint work with T. Kelly (Queen Mary University of London) on Chern characters of coherent sheaves on projective hypersurfaces. The formulas we get will be explicitly in terms of both the terms and differentials in the infinite free resolution of the corresponding graded module. We will also discuss some examples like complete intersections and applications to special cases of the Hodge conjecture.
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