Abstract
LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
[An] Andreatta, M., Contraction of Gorenstein polarized varieties with high nef value, Math. Ann.300 (1994) 669–679.
[A-S] Andreatta, M.—Sommese A.J., Generically ample divisors on normal Gorenstein surfaces, in Singularities—Contemporary Math.90 (1989), 1–20.
[A-W1] Andreatta, M.—Wiśniewski, J.: A note on non vanishing and its applications, Duke Math. J.72 (1993).
[A-W2] Andreatta, M.—Wiśniewski, On good contractions of smooth of smooth 4-folds, in preparation.
[B-S] Beltrametti, M.—Sommese, A.J., On the adjunction theoretic classification of polarized varieties, J. reine und angew. Math.427 (1992), 157–192.
[E-W] Esnault, H., Vieheweg E.Lectures on Vanishing Theorems, DMV Seminar Band 20, Birkhäuser Verlag (1992)
[Fu1] Fujita, T., On polarized Manifolds whose adjoint bundles are not semipositive, Advanced Studies in Pure Mathematics10 (1987), Algebraic Geometry Sendai 1985, 167–178.
[Fu2] Fujita, T., Classification theories of polarized varieties, London Lect. Notes115, Cambridge Press 1990.
[Fu3] Fujita, T., Remarks on quasi-polarized varieties, Nagoya Math. J.115 (1989), 105–123.
[Ha] Harshorne, R.,Algebraic Geometry, Springer-Verlag, 1977.
[Ma] Maeda, H. Ramification divisors for branched coverings ofP n, Math. Ann.288 (1990), 195–198.
[L-S] Lipman, J.—Sommese, A.J., On blowing down projective spaces in singular varieties, J. reine und angew. Math.362 (1985), 51–62.
[K-M-M] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the Minimal Model Program inAlgebraic Geometry, Sendai, Adv. Studies in Pure Math.10, Kinokuniya-North-Holland 1987, 283–360.
[K-O] Kobayashi, S.—Ochiai, T., On complex manifolds with positive tangent bundles, J. Math. Soc. Japan22 (1970), 499–525.
[K-M] Kollár, J.—Mori, S., Classification of three-dimensional flips, Journal of the A.M.S.,5 (1992), 533–703.
[Sa] Sakai, F., Ample Cartier divisors on normal surfaces, J. reine und angew. Math.366 (1986), 121–128.
[Y-Z] Ye, Y.G.—Zhang, Q., On ample vector bundle whose adjunction bundles are not numerically effective, Duke Math. Journal,60, n. 3 (1990), p. 671–687.