Complexity of del Pezzo surfaces with du Val singularities
Valentine Nadler
TL;DR
The work addresses how far del Pezzo surfaces with du Val singularities are from toric by computing the complexity γ(X) through the σ invariant, using minimal resolutions to convert the problem to smooth weak del Pezzo settings and analyzing boundary divisors on cycles in D(Y). The approach combines cycle-based induction on degree, careful treatment of snc and non-snc configurations, and a tree-structure algorithm to determine σ′(Y) and hence γ(X) across degrees d=1,...,9. The main results give explicit γ(X) and σ(X) ranges depending on d (e.g., γ(X)=0 for d≥7; γ(X)∈{0,1} for d∈{5,6}; detailed σ(X) values for smaller d) and provide a comprehensive framework for handling various geometric configurations on resolutions. These findings sharpen our understanding of toric proximity in singular del Pezzo surfaces and offer concrete geometric conditions that control the complexity via boundary divisors and cycle structures on resolutions.
Abstract
We compute the complexity of del Pezzo surfaces with du Val singularities.