Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-19T17:34:19.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized adjunction and applications

Published online by Cambridge University Press:  24 October 2008

Paltin Ionescu
Paltin Ionescu
Affiliation:
Department of Mathematics, University of Bucharest, 70109 Bucharest, Romania
Get access Rights & Permissions [Opens in a new window]

Extract

The linear system |K + C| ‘adjoint’ to a curve C on a projective surface was studied by the classical Italian geometers. The adjoint system to a hyperplane section H of smooth projective surface was investigated systematically, in modern terms, by Sommese [22] and Van de Ven [26]. The map associated to the linear system |K + (r−1)H|, where H is a hyperplane section of a smooth variety of arbitrary dimension r, was used to classify submanifolds of ℙn with ‘small invariants’ (e.g. degree, sectional genus, etc.); see [10]. On the other hand, Sommese [23, 24, 25] studied adjoint systems to a smooth ample divisor H on a smooth threefold X and obtained, as applications, many interesting results about the pair (X, H). As noticed independently by several authors (see e.g. [17], [4], [11]) the appearance of Mori's deep contribution [20] (see also [21]) put the subject of adjunction in a new perspective. Accordingly, the present paper–which relies heavily on Mori's results and on the contraction theorem due to Kawamata-Shokurov (see [14])–contains a systematical study of various adjoint systems to an ample (possibly non-effective) divisor on a manifold of arbitrary dimension. More precisely, the main result (which is contained in Section 1) gives the precise description of polarized pairs (X, H), where X is a complex projective mani–fold of dimension r and H an ample divisor on it (not necessarily effective), such that Kx + iH is not semiample (respectively ample) for 1 ≤ i = r + 1, r, r − 1, r − 2 (respectively i = r + 1, r, r − 1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 457 - 472
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]BӐdescu, L.. On ample divisors. I. Nagoya Math. J. 86 (1982), 155–171.CrossRefGoogle Scholar
[2]BӐdescu, L.. On ample divisors. II. In Proceedings of the week of Algebraic Geometry, Bucharest 1980, Teubner-Texte zur Mathematik, Bank 40 (Leipzig, 1981).Google Scholar
[3]BӐdescu, L.. The projective plane blown-up at a point as an ample divisor. Atti Accad. Ligure Sci. Lett. 38 (1981), 3–7.Google Scholar
[4]Beltrametti, M. and Pallesohi, M.. On threefolds with low sectional genus. Nagoya Math. J. 101 (1986), to appear.CrossRefGoogle Scholar
[5]Ein, L.. The ramification divisor for branched coverings of . Math. Ann. 261 (1982), 483–485.CrossRefGoogle Scholar
[6]Fujita, T.. On the structure of polarized varieties with Δ-genera zero. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103–115.Google Scholar
[7]Fujita, T.. On the hyperplane section principle of Lefscheftz. j. Math. Soc. Japan 32 (1980), 153–169.Google Scholar
[8]Fujita, T.. On the structure of polarized manifolds with total deficiency one. I, II and III. j. Math. Soc. Japan 32 (1980), 709–725; 33 (1981) 415–434; 36 (1984) 75–89.Google Scholar
[9]Hartshorne, R.. Algebraic Geometry (Springer-Verlag, 1977).CrossRefGoogle Scholar
[10]Ionescu, P.. Embedded projective varieties of small invariants. In Proceedings of the week of Algebraic Geometry, Bucharest 1982, Springer Lecture Notes in Math., vol. 1056 (1984).Google Scholar
[11]Ionescu, P.. On varietics whose degree is small with respect to codimension. Math. Ann. 271 (1985), 339–348.CrossRefGoogle Scholar
[12]Iskovskih, V. A.. Fano 3-folds. I. (Translated by M. Reid) math. USSR I zv. 11 (1977), 485–527.CrossRefGoogle Scholar
[13]Kawamata, Y.. A generalization of kodaira-Ramanujam’s vanishing theorem. Math. Ann. 261 (1982), 43–46.CrossRefGoogle Scholar
[14]Kawamata, Y.. The cone of curves of algebraic varieties. Ann. Math. 119 (1984), 603–633.CrossRefGoogle Scholar
[15]Kleiman, S. L.. Toward a numerical theory of ampleness. Ann. Math. 84 (1986), 293–344.CrossRefGoogle Scholar
[16]Kobayashi, S. and Ochiai, T.. Characterizations of complex projective spaces and hyper-quadrics. j. Math. Kyoto Univ. 13 (1973), 31–47.Google Scholar
[17]Lanteri, A. and Palleschi, M.About the adjunction process for polarized algebraic surfaces. J. reine angew. Math. 352 (1984), 15–23.Google Scholar
[18]Lanteri, A. and Palleschi, M.On the ampleness of Kx ⊗ Ln for a polarized threefold (X, L). Atti. Accad. Naz. Lincei. Rend. (8), 78 (1985), to appear.Google Scholar
[19]Mori, S.. Threefolds whose canonical bundles are not numerically effective. Ann. Math.. 116 (1982), 133–176.CrossRefGoogle Scholar
[20]Mori, S.. Projective manifolds with ample tangent bundles. Ann. Math. 110 (1979), 593–606.CrossRefGoogle Scholar
[21]Mori, S. Threefolds whose canonical bundles are not numerically effective. In In Algebraic Threefolds, proceedings Varenna 1981, Springer Lecture Notes in Math., vol. 947 (1982).Google Scholar
[22]Sommese, A. J.. Hyperplane sections of projective surfaces. I. The adjunction mapping. Duke Math. J. 46 (1979), 377–401.CrossRefGoogle Scholar
[23]Sommese, A. J.On the minimality of hyperplane sections of projectie threefolds. J. reine angew. Math 329 (1981), 16–41.Google Scholar
[24]Sommese, A. J.. The birational theory of hyperplane sections of projective threefolds (Preprint, 1981).Google Scholar
[25]Sommese, A. J.. Ample divisors on 3-folds. In Algebraic Threefolds, Proceedings Varenna 1981, Springer Lecture Ntoes in Math. Vol. 947 (1982).Google Scholar
[26]Vandeven, A.. On the 2-connectedness of very ample divisors on a surface. Duke Math. J. 46 (1979), 403–407.Google Scholar
[27]Viehweg, E.. vanishing theorems. J. reine angew. Math. 335 (1982), 1–8.Google Scholar
[28]Wahl, J. M.. A cohomological characterization of ℙn. Inv math. 72 (1993), 351–322.Google Scholar