Lahyane, Mustapha Exceptional curves on smooth rational surfaces with \(-K\) not nef and of self-intersection zero. (English) Zbl 1069.14041 Proc. Am. Math. Soc. 133, No. 6, 1593-1599 (2005). 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. Reviewer: Tommaso de Fernex (Ann Arbor) Cited in 3 Documents MSC: 14J26 Rational and ruled surfaces 14E05 Rational and birational maps 14C20 Divisors, linear systems, invertible sheaves Keywords:anticanonical rational surfaces; minimal models of smooth rational surfaces; Hodge index theorem; points in general position; Néron-Severi group; blowing-up Citations:Zbl 0652.14003; Zbl 1062.14046 PDFBibTeX XMLCite \textit{M. Lahyane}, Proc. Am. Math. Soc. 133, No. 6, 1593--1599 (2005; Zbl 1069.14041) Full Text: DOI References: [1] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023 [2] Brian Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1191 – 1208. · Zbl 0860.14006 [3] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [4] Mustapha Lahyane, Rational surfaces having only a finite number of exceptional curves, Math. Z. 247 (2004), no. 1, 213 – 221. · Zbl 1062.14046 · doi:10.1007/s00209-002-0474-y [5] Jeffrey A. Rosoff, Effective divisor classes and blowings-up of \?², Pacific J. Math. 89 (1980), no. 2, 419 – 429. · Zbl 0564.14002 [6] Rick Miranda and Ulf Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), no. 4, 537 – 558. · Zbl 0652.14003 · doi:10.1007/BF01160474 [7] Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 271 – 293. · Zbl 0100.16801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.