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Singular $\mathbb{Q}$-homology planes of negative Kodaira dimension have smooth locus of non-general type

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  • Mathematics

Abstract

Palka, K. and Koras, M. Osaka J. Math. 50 (2013), 61–114 SINGULAR Q-HOMOLOGY PLANES OF NEGATIVE KODAIRA DIMENSION HAVE SMOOTH LOCUS OF NON-GENERAL TYPE KAROL PALKA and MARIUSZ KORAS (Received May 11, 2011) Abstract We show that if a normal Q-acyclic complex surface has negative Kodaira dimen- sion then its smooth locus is not of general type. This generalizes an earlier result of Koras–Russell for contractible surfaces. Contents 1. Main result ..... . .. . .. .. . .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 61 2. Notation and preliminaries ....... . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 63 3. Basic properties and some inequalities ..... . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 70 4. Bounding the shape of the exceptional divisor ...... . .. .. . .. . .. .. . .. . .. .. . . 74 5. Special affine rulings of the resolution ...... .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 80 6. The boundary is a fork ...... .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 87 7. Some intermediate surface containing the smooth locus ....... . .. . .. .. . . 95 8. Special cases ...... .. .. . .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 108 References ....... . .. .. . .. . .. .. . .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . .. . .. .. . . 112 1. Main result We work in the category of complex algebraic varieties. We continue the program of classification of Q-homology planes. A normal surface S0 is called a Q-homology plane if its rational cohomology is the same as that of the affine plane C2, i.e. H�(S0,Q) Q. Properties of these surfaces have been analyzed for a long time, motivations come from studies on the cancellation conjecture of Zariski, on the two-dimensional Jacobian con- jecture, on quotients of actions of reductive groups on affine spaces or on ex

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