Cylinders in singular del Pezzo surfaces. (21st April 2016)

• Record Type: Journal Article
• Authors: Cheltsov, Ivan; Park, Jihun; Won, Joonyeong
• Is Part Of: Compositio mathematica. Volume 152:Number 6(2016:Nov.)
• Page Start: 1198
• Page End: 1224
• Publication Date: 2016-04-21
• Subjects: 14C20, -- 14E05, -- 14J17, -- 14J26, -- 14J45 (primary), -- 14R20, -- 14R25 (secondary); affine cone, -- anticanonical divisor, -- cylinder, -- del Pezzo surface, -- du Val singularity, -- Ga -action, -- minimal resolution, -- log canonical singularity, -- weak del Pezzo surface; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X16007284
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$ -actions on their affine cones defined by their anticanonical divisors.