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On the adjunction theoretic classification of projective varieties. (English) Zbl 0712.14029
Let $$X^{\wedge}$$ be a connected projective submanifold of $${\mathbb{P}}_{{\mathbb{C}}}$$ of dimension $$n\geq 3$$, and let $$L^{\wedge}={\mathcal O}_{{\mathbb{P}}_{{\mathbb{C}}}}(1)_{X^{\wedge}}$$. In this article we give a precise biregular structure theory for those pairs $$(X^{\wedge},L^{\wedge})$$ with $$n\geq 6$$, such that $$| m(K_{X^{\wedge}}+(n-3)L^{\wedge})|$$ does not give a birational map for any positive $$m.$$ For $$n=3, 4, 5$$, we give partial results, i.e. we give a precise biregular structure theory for those pairs $$(X^{\wedge},L^{\wedge})$$ such that $$| m(K_{X^{\wedge}}+tL^{\wedge})|$$ does not give a birational map for any positive m with mt integral and $$t=2/3, 3/2$$, and 9/4 for $$n=3, 4, 5$$ respectively. In the case of $$n=4$$ these results were obtained by M. L. Fania and the reviewer [Ark. Mat. 27, No.2, 245-256 (1989; Zbl 0703.14005)].
Some simple applications of these results are given. Let $$(X^{\wedge},L^{\wedge})$$ be such that $$n\geq 6$$, and assume that the degree of $$X^{\wedge}$$ embedded by $$| L^{\wedge}|$$ is $$>g(L^{\wedge})-1,$$ where $$g(L^{\wedge})$$ is the genus of a curve section of $$X^{\wedge}$$. Then $$| m(K_{X^{\wedge}}+(n- 3)L^{\wedge})|$$ does not give a birational map for any positive $$m,$$ and we can apply our result to get precise information on the biregular structure for these pairs. Some applications are also made to pairs $$(X^{\wedge},L^{\wedge})$$ with the discriminant locus, $${\mathcal D}$$, not a divisor on $$| L^{\wedge}|$$.
Reviewer: A.Sommese

##### MSC:
 14N05 Projective techniques in algebraic geometry 14M99 Special varieties
##### Keywords:
adjunction; classification of projective varieties
Full Text:
##### References:
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