Juan Pablo Zuñiga
Pontificia Universidad Católica de Chile
Title
Wahl singularities in degenerations of del Pezzo surfaces
Abstract
Motivated by the classification of Q-Gorenstein degenerations of the projective plane with Wahl singularities by Hacking and Prokhorov (2010), we study which Wahl singularities can arise in Q-Gorenstein degenerations of del Pezzo surfaces. We give a complete description of all Wahl singularities appearing in degenerations of del Pezzo surfaces of degree d, for any fixed 1<=d<=9. To this end, we introduce del Pezzo Wahl chains with markings, which define marked del Pezzo surfaces. These surfaces dominate all possible degenerations of del Pezzo surfaces with a unique Wahl singularity and big anticanonical divisor. Moreover, they admit canonical toric degenerations with only T-singularities and are in one-to-one correspondence with certain fake weighted projective planes, extending the picture known for degree 9. I will also discuss a connection with recent results of Polishchuk and Rains on exceptional vector bundles, viewed through Hacking’s exceptional collections, a tool for constructing exceptional collections of vector bundles via Q-Gorenstein smoothings of surfaces with at most Wahl singularities. This is joint work with Giancarlo Urzúa.