Let X be a smooth complex projective variety and let Z = (s = 0) be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dim Z = dim X - r. We show with some examples that in general the Kleiman-Mori cones NE(Z) and NE(X) are different. We then give a necessary and sufficient condition for an extremal ray in NE(X) to be also extremal in NE(Z). We apply this result to the case r = 1 and Z a Fano manifold of high index; in particular we classify all X with an ample divisor which is a Mukai manifold of dimension >= 4. In the last section we prove a general result in case Z is a minimal variety with 0 <= n(Z) < dim Z. (C) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Andreatta, M., Novelli, C., Occhetta, G. (2006). Connections between the geometry of a projective variety and of an ample section. MATHEMATISCHE NACHRICHTEN, 279(13-14), 1387-1395 [10.1002/mana.200410427].
Connections between the geometry of a projective variety and of an ample section
NOVELLI, CARLA;
2006
Abstract
Let X be a smooth complex projective variety and let Z = (s = 0) be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dim Z = dim X - r. We show with some examples that in general the Kleiman-Mori cones NE(Z) and NE(X) are different. We then give a necessary and sufficient condition for an extremal ray in NE(X) to be also extremal in NE(Z). We apply this result to the case r = 1 and Z a Fano manifold of high index; in particular we classify all X with an ample divisor which is a Mukai manifold of dimension >= 4. In the last section we prove a general result in case Z is a minimal variety with 0 <= n(Z) < dim Z. (C) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.