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Cylinders in singular del Pezzo surfaces
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 6 / June 2016
- Published online by Cambridge University Press:
- 21 April 2016, pp. 1198-1224
- Print publication:
- June 2016
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- Article
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- You have access
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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.
