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Projective manifolds with small pluridegrees

Authors:	Mauro C Beltrametti Andrew J Sommese

Institution:	Dipartimento di Matematica, Università Degli Studi di Genova, Via Dodecaneso 35, I-16146 Genova, Italy ; Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

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 at a finite set , 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, , where is the genus of a curve section of $(\mathcal{M},\mathcal{L})$ , and 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 $\|2(K_\mathcal{M}+ (n-2)\mathcal{L})\|$ .

Keywords:	Smooth complex polarized $n$-fold very ample line bundle adjunction theory log-general type pluridegrees

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