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 \({\mathbb{P}}^ 3\).

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 |