Fix integers n, r and d such that 2 ≤ n ≤ r − 1 and let V(n, r, d) be the set of all smooth, irreducible, projective, and nondegenerate varieties V of degree d and dimension n in the projective space P r , such that, only for n ≥ 3, the canonical bundle K V is numerically effective (i.e. K V ·C ≥ 0 for all curves C ⊂ V). Put K(n, r, d) = sup {K V n : V ε ν (n, r, d)}. In this paper we compute K(n, r, d) in terms of n, r and d and classify the varieties attaining the bound, at least when d ≫ r and for some range of n and r. In the case of surfaces, i.e. when n = 2, in the case r ≥ 2n + 1 and for d ≫ r we give a complete classification. It turns out that only in certain cases the varieties reaching the bound K (n, r, d) are Castelnuovo's varieties, that is varieties of maximal geometric genus.

DI GENNARO, V. (2001). Self-intersection of the canonical bundle of a projective variety. COMMUNICATIONS IN ALGEBRA, 29(1), 141-156 [10.1081/AGB-100000790].

Self-intersection of the canonical bundle of a projective variety

DI GENNARO, VINCENZO
2001-01-01

Abstract

Fix integers n, r and d such that 2 ≤ n ≤ r − 1 and let V(n, r, d) be the set of all smooth, irreducible, projective, and nondegenerate varieties V of degree d and dimension n in the projective space P r , such that, only for n ≥ 3, the canonical bundle K V is numerically effective (i.e. K V ·C ≥ 0 for all curves C ⊂ V). Put K(n, r, d) = sup {K V n : V ε ν (n, r, d)}. In this paper we compute K(n, r, d) in terms of n, r and d and classify the varieties attaining the bound, at least when d ≫ r and for some range of n and r. In the case of surfaces, i.e. when n = 2, in the case r ≥ 2n + 1 and for d ≫ r we give a complete classification. It turns out that only in certain cases the varieties reaching the bound K (n, r, d) are Castelnuovo's varieties, that is varieties of maximal geometric genus.
2001
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03 - GEOMETRIA
English
DI GENNARO, V. (2001). Self-intersection of the canonical bundle of a projective variety. COMMUNICATIONS IN ALGEBRA, 29(1), 141-156 [10.1081/AGB-100000790].
DI GENNARO, V
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/91024
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