Exceptional curves on smooth rational surfaces with -𝐾 not nef and of self-intersection zero

Lahyane, Mustapha

doi:10.1090/S0002-9939-04-07693-2

Exceptional curves on smooth rational surfaces with $-K$ not nef and of self-intersection zero
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Proc. Amer. Math. Soc. 133 (2005), 1593-1599 Request permission

Abstract:

A $(-n)$-curve is a smooth rational curve of self-intersection $-n$, where $n$ is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface $X$ defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has $(-2)$-curves. In this paper we prove that for such a surface $X$, the set of $(-1)$-curves on $X$ is finite but non-empty, and that $X$ may have no $(-2)$-curves. Related facts are also considered.

References

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. MR 749574, DOI 10.1007/978-3-642-96754-2
Brian Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1191–1208. MR 1373636, DOI 10.1090/S0002-9947-97-01722-4
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Mustapha Lahyane, Rational surfaces having only a finite number of exceptional curves, Math. Z. 247 (2004), no. 1, 213–221. MR 2054527, DOI 10.1007/s00209-002-0474-y
Jeffrey A. Rosoff, Effective divisor classes and blowings-up of $\textbf {P}^{2}$, Pacific J. Math. 89 (1980), no. 2, 419–429. MR 599129
Rick Miranda and Ulf Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), no. 4, 537–558. MR 867347, DOI 10.1007/BF01160474
Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 271–293. MR 126444, DOI 10.1215/kjm/1250775912