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Remarks on quasi-polarized varieties

Published online by Cambridge University Press:  22 January 2016

Takao Fujita
Takao Fujita*
Affiliation:
Department of Mathematics, College of Arts and Sciences University of Tokyo, Komaba, Meguro, Tokyo 153, Japan
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Let V be a variety, which means, an irreducible reduced projective scheme over an algebraically closed field of any characteristic. A line bundle L on V is said to be nef if LC ≧ 0 for any curve C in V. Thus, “nef” is never an abbreviation of “numerically equivalent to an effective divisor”. L is said to be big if k(L) = n = dim V. In case L is nef, it is big if and only if Ln > 0 (cf. [F7; (6.5)]. When L is nef and big, the pair (V, L) will be called a quasi-polarized variety.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 115 , September 1989 , pp. 105 - 123
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[F1] Fujita, T., On the structure of polarized varieties with Δ-genera zero, J. Fac. Sci. Univ. of Tokyo, 22 (1975), 103–115.Google Scholar
[F2] Fujita, T., On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan, 32 (1980), 153–169.CrossRefGoogle Scholar
[F3] Fujita, T., On the structure of polarized manifolds with total deficiency one, part I, II & III, J. Math. Soc. Japan, 32 (1980), 709–725, 33 (1981), 415–434 & 36 (1984), 75–89.Google Scholar
[F4] Fujita, T., On L-dimension of coherent sheaves (with correction), J. Fac. Sci. Univ. of Tokyo, 28 (1981), 215–236 & 29 (1982), 719–720.Google Scholar
[F5] Fujita, T., On polarized varieties of small Δ-genera, Tôhoku Math. J., 34 (1982), 319–341.Google Scholar
[F6] Fujita, T., Theorems of Bertini type for certain types of polarized manifolds, J. Math. Soc. Japan, 34 (1982), 709–718.Google Scholar
[F7] Fujita, T., Semipositive line bundles, J. Fac. Sci. Univ. of Tokyo, 30 (1983), 353–378.Google Scholar
[F8] Fujita, T., Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan, 38 (1986), 19–37.Google Scholar
[F9] Fujita, T., Projective varieties of Δ-genus one, in Algebraic and Topological Theories—to the memory of Dr. Takehiko MIYATA, pp. 149–175, Kinokuniya, Tokyo, 1985.Google Scholar
[F10] Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive, in Algebraic Geometry Sendai 1985, pp. 167–178, Advanced Studies in Pure Math. 10, Kinokuniya, 1987.Google Scholar
[F11] Fujita, T., Notes on quasi-polarized varieties, Proc. Japan Acad., 64 (1988), 88–90.Google Scholar
[H] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann of Math., 79 (1964), 109–326.Google Scholar
[I] Ionescu, P., Generalized adjunction and applications, Math. Proc. Cambridge Phil. Soc, 99 (1986), 457–472.Google Scholar
[K] Kawamata, Y., A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261 (1982), 43–46.Google Scholar
[KMM] Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic Geometry Sendai 1985, pp. 283–360, Advanced Studies in Pure Math. 10, Kinokuniya, 1987.Google Scholar
[M1] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. Math., 116 (1982), 133–176.Google Scholar
[M2] Mori, S., Flip theorem and the existence of minimal models for threefolds, J. of AMS, 1 (1988), 117–253.Google Scholar
[R] Reid, M., Canonical 3-folds, in Géométrie Algébrique Angers 1979 pp. 273–310, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.Google Scholar
[Sa] Sakai, F., Ample Cartier divisors on normal surfaces, J. reine angew. Math., 366 (1986), 121–128.Google Scholar
[Sh] Shokurov, V. V., Theorem on non-vanishing, Math. USSR-Izv., 26 (1986), 591–604.Google Scholar
[S1] Sommese, A. J., On manifolds that cannot be ample divisors, Math. Ann., 221 (1976), 55–72.Google Scholar
[S2] Sommese, A. J., On the adjunction theoretic structure of projective varieties, in Complex Analysis and Algebraic Geometry, pp. 175–213, Lecture Notes in 1194, Springer, 1986.Google Scholar
[U] Ueno, K., Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Math. 439, Springer, 1975.Google Scholar
[V] Viehweg, E., Vanishing theorems, J. reine angew. Math., 335 (1982), 1–8.Google Scholar