Objects of a Phantom on a Rational Surface
Amal Mattoo
TL;DR
The paper investigates phantom subcategories inside the derived category of coherent sheaves on $X=\text{Bl}_{10\text{pts}}\mathbf{P}^{2}$, aiming to extract geometric information from the phantom by constructing explicit objects. It develops projection techniques via the left adjoint $i^{*}$ to the phantom embedding and a projection spectral sequence to compute Hom-spaces, culminating in three main constructions: a strong generator, projections of skyscraper sheaves, and a two-cohomology object family. The strong generator produces a co-connective dg-algebra whose derived category recovers the phantom, while the projected skyscrapers yield a birational map from $X$ to a component of the moduli of objects, encoding the blowdown to $\mathbf{P}^{2}$. The two-cohomology objects provide explicit, simple, universally gluable members whose deformation theory reflects both the curve $C$ and its embedding in $X$, revealing additional irreducible components of the moduli and suggesting intrinsic categorical characterizations within the phantom. Together, these results demonstrate how detailed object-level data within a phantom retains rich geometric and birational information about the ambient variety, advancing understanding of how additive invariants relate to geometry in derived categories.
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.