Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Projective manifolds with small pluridegrees

Authors: Mauro C. Beltrametti and Andrew J. Sommese
Journal: Trans. Amer. Math. Soc. 352 (2000), 3045-3064
MSC (1991): Primary 14J40; Secondary 14M99, 14C20
Published electronically: May 21, 1999
MathSciNet review: 1641087
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Abstract: Let $\mathcal{L}$ be a very ample line bundle on a connected complex projective manifold $\mathcal{M}$ of dimension $n\ge 3$. Except for a short list of degenerate pairs $(\mathcal{M},\mathcal{L})$, $\kappa(K_\mathcal{M}+(n-2)\mathcal{L})=n$ and there exists a morphism $\pi: \mathcal{M} \to M$ expressing $\mathcal{M}$ as the blowup of a projective manifold $M$ at a finite set $B$, with $\mathcal{K}_M:=K_M+(n-2)L$ nef and big for the ample line bundle $L:= (\pi _*\mathcal{L})^{**}$. The projective geometry of $(\mathcal{M},\mathcal{L})$ is largely controlled by the pluridegrees $d_j:=L^{n-j}\cdot (K_M+(n-2)L)^j$ for $j=0,\ldots,n$, of $(\mathcal{M},\mathcal{L})$. For example, $d_0+d_1=2g-2$, where $g$ is the genus of a curve section of $(\mathcal{M},\mathcal{L})$, and $d_2$ is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of $(\mathcal{M},\mathcal{L})$. In this article, a detailed analysis is made of the pluridegrees of $(\mathcal{M},\mathcal{L})$. The restrictions found are used to give a new lower bound for the dimension of the space of sections of $\mathcal{K}_M$. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to $|2(K_\mathcal{M}+ (n-2)\mathcal{L})|$.

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Additional Information

Mauro C. Beltrametti
Affiliation: Dipartimento di Matematica, Università Degli Studi di Genova, Via Dodecaneso 35, I-16146 Genova, Italy

Andrew J. Sommese
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Keywords: Smooth complex polarized $n$-fold, very ample line bundle, adjunction theory, log-general type, pluridegrees.
Received by editor(s): February 8, 1998
Published electronically: May 21, 1999
Article copyright: © Copyright 2000 American Mathematical Society