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On polarized normal varieties, I

Published online by Cambridge University Press:  22 January 2016

T. Matsusaka
T. Matsusaka*
Affiliation:
Department of Mathematics, Brandeis University, Waltham, Massachusetts 02154, U.S.A.
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In this paper, we deal with the first part of an application of our Riemann-Roch type inequalities (cf. [13], [23]) toward deformations of polarized normal varieties. In Chapter I, we discuss the problem of eliminating fixed components from complete linear systems defined by multiples of a given divisor. Let Un be a complete normal variety and Y0 an ample Cartier divisor on U. The main result of the chapter is that there is a positive integer c0, predicted by the first two leading coefficients of the polynomial X(U, O(m Y0)), such that the complete linear system Λ(rc0Y0) has no fixed component whenever r is a positive integer (which is essentially contained in Theorem 1.1 (cf. [13], Lemma 5.2)). An easy consequence of this result is that when n = 2, we can find another positive integer c1 predicted by the same coefficients as above, such that rc1c0Y0 is very ample on U whenever r is a positive integer. Even though this has been generalized to n = 3 by J. Kollár (cf. [12]), we have included this in Section 3 since it is very simple.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 175 - 211
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Artin, M., Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math., 84 (1962), 485–496.CrossRefGoogle Scholar
[ 2 ] Artin, M., Isolated rational singularities of surfaces, Amer. J. Math., 88 (1966), 129–136.Google Scholar
[ 3 ] Chow, W. L. and Waerden, v. d., Zur Algebraischen Geometrie IX, Math. Ann., 113 (1937), 692–704.CrossRefGoogle Scholar
[ 4 ] Elkik, R., Singularités rationelles et deformations, Invent. Math., 47 (1976), 139–147.Google Scholar
[ 5 ] Grothendieck, A. and Dieudonne, J., Ùléments de géometrie algébrique, Publ. Math. IHES, Paris.Google Scholar
[ 6 ] Grauert, H. and Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf Komplexen Raumen, Invent. Math., 11 (1970), 262–292.CrossRefGoogle Scholar
[ 7 ] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, I—II, Ann. of Math., 79 (1964), 109–326.Google Scholar
[ 8 ] Kawamata, Y., A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261 (1982), 43–46.CrossRefGoogle Scholar
[ 9 ] Kempf, G., Toroidal embeddings, I, Springer Lecture Notes, 339 (1973), 41–52.Google Scholar
[10] Kleiman, S., Toward a numerical theory of ampleness, Ann. of Math., 84 (1966), 293–344.CrossRefGoogle Scholar
[11] Kodaira, K., On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci., U.S.A., 39 (1953), 1268–1273.CrossRefGoogle ScholarPubMed
[12] Kollár, J., Toward moduli of singular varieties, Thesis at Brandeis, 1983, to appear soon.Google Scholar
[13] Kollár, J., and Matsusaka, T., Riemann-Roch type inequalities, Amer. J. Math., 105 (1983), 229–252.Google Scholar
[14] Lieberman, D. and Mumford, D., Matsusaka’s big theorem, Proc. Symp. Pure Math., 29 (1975), 513–530.CrossRefGoogle Scholar
[15] Matsusaka, T. and Mumford, D., Two fundamental theorems on deformations of polarized varieties, Amer. J. Math., 86 (1964), 668–684.Google Scholar
[16] Matsusaka, T., Polarized varieties with a given Hilbert polynomial, Amer. J. Math., 94 (1972), 1027–1076.Google Scholar
[17] Rosenlicht, M., Equivalence relations on algebraic curves, Ann. of Math., 56 (1952), 169–191.Google Scholar
[18] Shimura, G., Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. Math., 77 (1955), 134–176.Google Scholar
[19] Viehweg, E., Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1–8.Google Scholar
[20] Weil, A., Foundations of Algebraic Geometry, Amer. Math. Soc. Colloquium Publ., 29 (1960).Google Scholar
[21] Zariski, O., Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc, 53 (1943), 490–542.CrossRefGoogle Scholar
[22] Fujita, T., Semipositive line bundles, J. Fac. Sci., Univ. Tokyo, 30 (1983), 353–378.Google Scholar
[23] Matsusaka, T., A note and a correction to Riemann-Roch type inequalities, Amer. J. Math., 106 (1984), 1265–1268.Google Scholar
[24] Seidenberg, A., The hyperplane sections of normal varieties, Trans. Amer. Math. Soc, 69 (1950), 357–386.Google Scholar