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On the structure of polarized manifolds with total deficiency one. III. (English) Zbl 0541.14036
[For part I and II see ibid. 32, 709-725 (1980; Zbl 0474.14017) and 33, 415-434 (1981; Zbl 0474.14018).]
This is the third part of a series of author’s papers classifying nonsingular n-dimensional projective varieties V with an ample divisor L such that $$\dim | L| =L^ n+n-2.$$ These varieties are natural generalizations of Del Pezzo surfaces to higher dimension and represent a special class of Fano varieties (varieties with an ample canonical divisor). Let g(V,L) be the arithmetical genus of a divisor from the class $$L^{n-1}$$. In the case $$g(V,L)=1$$ all such varieties with $$L^ n\neq 1,5$$ (resp. $$=5)$$ were classified in part I of this paper (resp. in part II). In the present paper the author finishes this classification by considering the case $$L^ n=1$$. In this case these varieties turn out to be true analogues of Del Pezzo surfaces of degree 1: The linear system $$| L|$$ has one isolated base point and after blowing it up $$| L|$$ defines a flat morphism to $${\mathbb{P}}^{n-1}$$ whose fibres are reduced irreducible curves of arithmetical genus 1. Also there is a familiar double plane construction: V is isomorphic to a double cover of a cone over the Veronese variety $$v_ 2({\mathbb{P}}^{n-1})$$ branched along a nonsingular divisor cut by a hypersurface of degree 3.
The case $$g=g(V,L)>1$$ is considered also, but only under the assumption that the general member of $$L^{n-1}$$ can be represented by a hyperelliptic curve of genus g(V,L). The varieties with this property are divided into 3 classes, (-), ($$\infty)$$ and $$(+)$$. The varieties of the first type are isomorphic to a hypersurface of degree $$4g+2$$ in a weighted projective space $${\mathbb{P}}(2g+1,2,1,...,1)$$. The varieties of the second class are isomorphic to the blowing-up at one point of a double cover of $${\mathbb{P}}^ n$$ ($$n\leq g)$$ branched along a nonsingular hypersurface of degree $$2g+2$$. Finally, the varieties of the third type are rational surfaces isomorphic to the blowing-up at one point of a double cover of a quadric branched along a nonsingular curve of bidegree $$(2g+2,2).$$
As an application, the author proves that every Kähler threefold of the same cohomological type as $${\mathbb{P}}^ 3$$ with positive $$c_ 1$$ must be isomorphic to $${\mathbb{P}}^ 3$$.
Reviewer: I.V.Dolgachev

##### MSC:
 14J40 $$n$$-folds ($$n>4$$) 14C20 Divisors, linear systems, invertible sheaves 14J10 Families, moduli, classification: algebraic theory 14M99 Special varieties
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