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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the second adjunction mapping.
The case of a 1-dimensional image


Authors: Mauro C. Beltrametti and Andrew J. Sommese
Journal: Trans. Amer. Math. Soc. 349 (1997), 3277-3302
MSC (1991): Primary 14E35, 14C20, 14J40
MathSciNet review: 1401513
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Abstract: Let $\widehat {L}$ be a very ample line bundle on an $n$-dimensional projective manifold $\widehat {X}$, i.e., assume that $\widehat {L}\approx i^*{\mathcal O}_{{\mathbb P}^{N}}(1)$ for some embedding $i:\widehat {X}\hookrightarrow {\mathbb P}^{N}$. In this article, a study is made of the meromorphic map, $\widehat {\varphi } : \widehat {X}\to \Sigma $, associated to $|K_{\widehat {X}}+(n-2)\widehat {L}|$ in the case when the Kodaira dimension of $K_{\widehat {X}}+(n-2)\widehat {L}$ is $\ge 3$, and $\widehat {\varphi }$ has a $1$-dimensional image. Assume for simplicity that $n=3$. The first main result of the paper shows that $\widehat \varphi $ is a morphism if either $h^0(K_{\widehat X}+\widehat L)\geq 7$ or $\kappa (\widehat {X})\geq 0$. The second main result of this paper shows that if $\kappa (\widehat X)\ge 0$, then the genus, $g(f)$, of a fiber, $f$, of the map induced by $\widehat \varphi $ on hyperplane sections is $\leq 6$. Moreover, if $h^0(K_{\widehat X}+\widehat L)\ge 21$ then $g(f)\leq 5$, a connected component $F$ of a general fiber of $\widehat \varphi $ is either a $K3$ surface or the blowing up at one point of a $K3$ surface, and $h^1({\mathcal O}_{\widehat X})\le 1$. Finally the structure of the finite to one part of the Remmert-Stein factorization of $\widehat \varphi $ is worked out.


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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
Email: beltrame@dima.unige.it

Andrew J. Sommese
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: sommese.1@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01809-6
PII: S 0002-9947(97)01809-6
Received by editor(s): January 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society