A criterion for a smooth surface to be del Pezzo. (English) Zbl 0838.14034
From the introduction: In this paper we assume throughout that the ground field \(k\) is algebraically closed and of characteristic zero. A smooth projective variety \(X\) is called Fano if its anticanonical bundle \(- K_X\) is ample. A Fano variety of dimension 2 is called a del Pezzo surface. — F. Campana and T. Peternell posed the following problem [Math. Ann. 289, No. 1, 169–187 (1991; Zbl 0729.14032)]: Let \(X\) be a smooth projective variety. Assume that \((- K_X) C > 0\) for all integral curves \(C\) on \(X\). Is \(X\) Fano?
If \(\dim X = 1\), then the assertion is trivial. The purpose of this paper is to give an affirmative answer to the problem in the case when \(\dim X = 2\). The precise statement of our result is as follows:
Theorem. Let \(X\) be a smooth projective surface. Then \((- K_X) C > 0\) for all integral curves \(C\) on \(X\) if and only if \(X\) is del Pezzo.
If \(\dim X = 1\), then the assertion is trivial. The purpose of this paper is to give an affirmative answer to the problem in the case when \(\dim X = 2\). The precise statement of our result is as follows:
Theorem. Let \(X\) be a smooth projective surface. Then \((- K_X) C > 0\) for all integral curves \(C\) on \(X\) if and only if \(X\) is del Pezzo.
MSC:
| 14J45 | Fano varieties |
| 14J26 | Rational and ruled surfaces |
| 14C20 | Divisors, linear systems, invertible sheaves |
Citations:
Zbl 0729.14032References:
| [1] | DOI: 10.2307/2007087 · Zbl 0544.14009 · doi:10.2307/2007087 |
| [2] | DOI: 10.1007/BF01446566 · Zbl 0729.14032 · doi:10.1007/BF01446566 |
| [3] | Hartshorne, Inst. Hautes ?tudes Sci. Pubi. Math. 29 pp 63– (1966) |
| [4] | Hartshorne, Algebraic Geometry 52 (1977) · doi:10.1007/978-1-4757-3849-0 |
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