A note on degenerations of del Pezzo surfaces

TL;DR

This paper investigates the singularities of degenerations of del Pezzo surfaces, establishing bounds on the number of non-Du Val singularities relative to the Picard number, and explores cases with maximal non-Du Val points.

Contribution

It provides a bound on the number of non-Du Val singularities in Q-Gorenstein degenerations of del Pezzo surfaces and analyzes cases with near-maximal singularities.

Findings

Number of non-Du Val singularities ≤ ρ(X)+2

Degenerations with ρ(X)+2 and ρ(X)+1 non-Du Val points are characterized

Bounds are sharp and relate to the Picard number

Abstract

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated.