A smooth irreducible nondegenerate projective variety X⊂PN is said to be a conic connected manifold (CC-manifold, for short) if through two general points of X there passes an irreducible conic contained in X. CC-manifolds are in particular rationally connected. The main theorem of the paper under review states that linearly normal CC-manifolds are Fano manifolds with second Betti number b2≤2. More precisely, it is proved that if such a CC-manifold X has b2=1 then Pic(X) is generated by the hyperplane sections and the index is at least (dimX+1)/2 except for the case X=v2(PN), the Veronese variety, and such CC-manifolds with b2=2 are completely classified: they are the inner projections from a linear subspace of the Veronese variety v2(PN), or Segre products of two projective spaces and their hyperplane sections. A characterization of rationality via covering families of 1-cycles is also provided. For a point x on a projective variety, an x-covering family is a covering family of rational 1-cycles such that every 1-cycle from the family passes through x and a general member is smooth at x. An x-covering family is smooth if all its 1-cycles are smooth. An x-covering family is said to satisfy the infinitesimal uniqueness property if a general member of the family is uniquely determined by its tangent space at x. The characterization is the following: a projective variety X is rational if and only if for some x∈X, X admits a smooth x-covering family satisfying the infinitesimal uniqueness property. This may be interpreted as a generalization of the characterization of projective space due to K. Cho, Y. Miyaoka and N. I. Shepherd-Barron [in Higher dimensional birational geometry (Kyoto, 1997), 1--88, Adv. Stud. Pure Math., 35, Math. Soc. Japan, Tokyo, 2002; MR1929792 (2003m:14080)].

Conic-connected manifolds

IONESCU, Paltin;
2010

Abstract

A smooth irreducible nondegenerate projective variety X⊂PN is said to be a conic connected manifold (CC-manifold, for short) if through two general points of X there passes an irreducible conic contained in X. CC-manifolds are in particular rationally connected. The main theorem of the paper under review states that linearly normal CC-manifolds are Fano manifolds with second Betti number b2≤2. More precisely, it is proved that if such a CC-manifold X has b2=1 then Pic(X) is generated by the hyperplane sections and the index is at least (dimX+1)/2 except for the case X=v2(PN), the Veronese variety, and such CC-manifolds with b2=2 are completely classified: they are the inner projections from a linear subspace of the Veronese variety v2(PN), or Segre products of two projective spaces and their hyperplane sections. A characterization of rationality via covering families of 1-cycles is also provided. For a point x on a projective variety, an x-covering family is a covering family of rational 1-cycles such that every 1-cycle from the family passes through x and a general member is smooth at x. An x-covering family is smooth if all its 1-cycles are smooth. An x-covering family is said to satisfy the infinitesimal uniqueness property if a general member of the family is uniquely determined by its tangent space at x. The characterization is the following: a projective variety X is rational if and only if for some x∈X, X admits a smooth x-covering family satisfying the infinitesimal uniqueness property. This may be interpreted as a generalization of the characterization of projective space due to K. Cho, Y. Miyaoka and N. I. Shepherd-Barron [in Higher dimensional birational geometry (Kyoto, 1997), 1--88, Adv. Stud. Pure Math., 35, Math. Soc. Japan, Tokyo, 2002; MR1929792 (2003m:14080)].
2010
Ionescu, Paltin; Russo, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1509117
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