Cylinders in singular del Pezzo surfaces

Abstract

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_S$ and such that the open set $S\setminus \mathrm{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial ${\mathbb{G}}_a$-actions on their affine cones defined by their anticanonical divisors.

23 pages

References in corpus (1)

• Ga-actions on affine cones

Cited by in corpus (6)

• Super-rigid Affine Fano Varieties
• Automorphisms of algebraic varieties and infinite transitivity
• Cylinders in weak del Pezzo fibrations
• Cylinders in canonical del Pezzo fibrations
• Cylindrical ample divisors on Du Val del Pezzo surfaces
• Notes on cylinders in smooth projective surfaces