Two examples of surfaces with normal crossing singularities

János Kollár

doi:10.1007/s11425-010-4161-x

Two examples of surfaces with normal crossing singularities

János Kollár

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in III. 2.6.2 of Éléments de géometrie algébrique, 1961, and raised again by Viehweg.

Original language	English (US)
Pages (from-to)	1707-1712
Number of pages	6
Journal	Science China Mathematics
Volume	54
Issue number	8
DOIs	https://doi.org/10.1007/s11425-010-4161-x
State	Published - Aug 2011

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

ampleness criteria
canonical ring
singular surface

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10.1007/s11425-010-4161-x

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title = "Two examples of surfaces with normal crossing singularities",

abstract = "This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in III. 2.6.2 of {\'E}l{\'e}ments de g{\'e}ometrie alg{\'e}brique, 1961, and raised again by Viehweg.",

keywords = "ampleness criteria, canonical ring, singular surface",

author = "J{\'a}nos Koll{\'a}r",

note = "Funding Information: Acknowledgements I thank A. Corti for a long discussion concerning an earlier log-surface whose canonical ring was not finitely generated and B. Hassett, B. Totaro and the referee for several corrections and suggestions. Partial financial support was provided by the NSF under grant number DMS-0500198. Part of this work was done while the author visited MSRI and the University of Texas.",

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N1 - Funding Information: Acknowledgements I thank A. Corti for a long discussion concerning an earlier log-surface whose canonical ring was not finitely generated and B. Hassett, B. Totaro and the referee for several corrections and suggestions. Partial financial support was provided by the NSF under grant number DMS-0500198. Part of this work was done while the author visited MSRI and the University of Texas.

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N2 - This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in III. 2.6.2 of Éléments de géometrie algébrique, 1961, and raised again by Viehweg.

AB - This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in III. 2.6.2 of Éléments de géometrie algébrique, 1961, and raised again by Viehweg.

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KW - canonical ring

KW - singular surface

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Two examples of surfaces with normal crossing singularities

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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