×

zbMATH — the first resource for mathematics

On the length of an extremal rational curve. (English) Zbl 0751.14007
It is proved that the exceptional locus of an elementary contraction of an algebraic variety is covered by rational curves whose degrees with respect to the canonical divisor \(K\) are bounded by twice the dimension. The theorem is formulated in the log category so that it applies, e.g., to the case in which \(K\) is relatively trivial. In the course of the proof, a subadjunction formula is proved using a vanishing theorem of Kodaira type.
Reviewer: Y.Kawamata (Tokyo)

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14H99 Curves in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [I] Ionescu, P.: Generalized adjunction and applications. Math. Proc. Camb. Philos. Soc.99, 457-472 (1986) · Zbl 0619.14004 · doi:10.1017/S0305004100064409
[2] [Ka1] Kawamata, Y.: The cone of curves of algebraic varieties. Ann. Math.119, 603-633 (1984) · Zbl 0544.14009 · doi:10.2307/2007087
[3] [Ka2] Kawamata, Y.: Moderate degenerations of algebraic surfaces. Proc. Symp. Bayreuth 1990 (to appear)
[4] [KMM] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Oda, T. (ed.) Algebraic Geometry. Proc. Symp., Sendai 1985. (Adv. Stud. Pure Math. vol. 10, pp. 283-360) Tokyo: Kinokuniyo 1987 · Zbl 0672.14006
[5] [Ko] Kollár, J.: The cone theorem: Note to [Ka1]. Ann. Math.120, 1-5 (1984) · Zbl 0544.14010 · doi:10.2307/2007069
[6] [MM] Miyaoka, Y., Mori, S.: A numerical criterion of uniruledness. Ann. Math.124, 65-69 (1986) · Zbl 0606.14030 · doi:10.2307/1971387
[7] [M] Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math.116, 133-176 (1982) · Zbl 0557.14021 · doi:10.2307/2007050
[8] [N] Nakayama, N.: The lower semi-continuity of the plurigenera of complex varieties. In: Oda, T. (ed.) Algebraic Geometry. Proc. Symp., Sendai 1985. (Adv. Stud. Pure Math. vol. 10, pp. 551-590) Tokyo: Kinokuniya 1987
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.