Wahl singularities in degenerations of del Pezzo surfaces

Abstract

For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities are classified using the Markov equation. To achieve this, we introduce del Pezzo Wahl chains with markings. They define marked del Pezzo surfaces $W_{*m}$ that govern all such degenerations. We also prove that every marked del Pezzo surface degenerates into a canonically defined toric del Pezzo surface with only T-singularities. In addition, we establish a one-to-one correspondence between the $W_{*m}$ surfaces and certain fake weighted projective planes. As applications, we show that every Wahl singularity occurs for del Pezzo surfaces of degree $\leq 4$, but that there are infinitely many Wahl singularities that do not arise for degrees $\geq 5$. We also use Hacking's exceptional collections to provide geometric proofs of recent results by Polishchuk and Rains on exceptional vector bundles on del Pezzo surfaces.