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Rational surfaces with \(K^ 2>0\). (English) Zbl 0874.14025

Summary: The main but not all of the results in this paper concern rational surfaces \(X\) for which the self-intersection \(K_X^2\) of the anticanonical class \(-K_X\) is positive. In particular, it is shown that no superabundant numerically effective divisor classes occur on any smooth rational projective surface \(X\) with \(K_X^2>0\). As an application, it follows that any 8 or fewer (possibly infinitely near) points in the projective plane \(\mathbb{P}^2\) are in good position. This is not true for 9 points, and a characterization of the good position locus in this case is also given. Moreover, these results are put into the context of conjectures for generic blowings up of \(\mathbb{P}^2\). All results are proven over an algebraically closed field of arbitrary characteristic.

MSC:

14J26 Rational and ruled surfaces
14C20 Divisors, linear systems, invertible sheaves
13P99 Computational aspects and applications of commutative rings
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References:

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