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Cylinders in singular del Pezzo surfaces

Part of: Surfaces and higher-dimensional varieties Birational geometry Cycles and subschemes Affine geometry

Published online by Cambridge University Press:  21 April 2016

Ivan Cheltsov ,
Jihun Park  and
Joonyeong Won
Ivan Cheltsov
Affiliation:
School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 7 Vavilova Str., Moscow 117312, Russia email I.Cheltsov@ed.ac.uk
Jihun Park
Affiliation:
Center for Geometry and Physics, Institute for Basic Science (IBS), 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 37673, Korea Department of Mathematics, POSTECH, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 37673, Korea email wlog@postech.ac.kr
Joonyeong Won
Affiliation:
Algebraic Structure and its Applications Research Center, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 34143, Korea email leonwon@kias.re.kr
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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.

Keywords

affine coneanticanonical divisorcylinderdel Pezzo surfacedu Val singularityGa -actionminimal resolutionlog canonical singularityweak del Pezzo surface

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves 14E05: Rational and birational maps 14J17: Singularities 14J26: Rational and ruled surfaces 14J45: Fano varieties
Secondary: 14R20: Group actions on affine varieties 14R25: Affine fibrations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1198 - 1224
Copyright
© The Authors 2016 

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