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 Q-divisor D that is Q-linearly equivalent to -K-S and such that the open set S\Supp(D) is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial G(a)-actions on their affine cones defined by their anticanonical divisors.

All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic 0 throughout this article.