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Singular \(\mathbb{Q}\)-homology planes of negative Kodaira dimension have smooth locus of non-general type. (English) Zbl 1287.14031

Let \(X\) be a normal algebraic surface/\(\mathbb{C}\) such that \(H_i(X; \mathbb{Q})= 0\) for \(i>0\). These surfaces occur naturally in affine algebraic geometry and have been studied by several mathematicians. Let \(X\) be a resolution of singularities of \(X\).
In this paper the author prove the following important result.
Theorem. With \(X\) as above, assume that \(\overline\kappa(\widetilde X)= -\infty\). Then \(\overline\kappa(X- \text{Sing}\, X)< 2\).
Here \(\overline\kappa\) denotes the logarithmic Kodaira dimension introduced by S. Iitaka. When \(X\) is smooth and contractible this result was proved by M. Koras and P. Russell [Transform. Groups 12, No. 2, 293–340 (2007; Zbl 1124.14050)]. The proof in the singular case uses similar methods from the theory of non-complete algebraic surfaces [T. Fujita’s results, Miyaoka- Yau type inequality proved by Kobayashi-Nakamura-Sakai,…). This result will be very useful to the workers in this area of surface theory.

MSC:

14R05 Classification of affine varieties
14J17 Singularities of surfaces or higher-dimensional varieties
14J26 Rational and ruled surfaces

Citations:

Zbl 1124.14050
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Full Text: arXiv Euclid

References:

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