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Exceptional curves on smooth rational surfaces with \(-K\) not nef and of self-intersection zero. (English) Zbl 1069.14041

By blowing up nine points on \({\mathbb P}^2\) in suitable (infinitely near) position, the author constructs an example of a smooth complex rational surface not containing any \((-2)\)-curve and whose anticanonical class is not nef and has zero self-intersection. Then he proves that every smooth complex rational surface whose anticanonical class satisfies these two conditions contains only finitely many \((-1)\)-curves. This result is motivated by a question posed by A. Hirschowitz. In an earlier paper [Math. Z. 247, 213–221 (2004; Zbl 1062.14046)], the author proved that a smooth complex rational surface with nef anticanonical class of self-intersection zero has only finitely many \((-1)\)-curves if any only if it contains \((-2)\)-curves [see also R. Miranda and U. Persson, Math. Z. 193, 537–558 (1986; Zbl 0652.14003)]. In 1998, Hirschowitz asked whether the same characterization holds in the case where \(-K_X\) is not nef; the result of the paper under review gives a negative answer to this question.

MSC:

14J26 Rational and ruled surfaces
14E05 Rational and birational maps
14C20 Divisors, linear systems, invertible sheaves
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