Let X be a smooth complex projective variety and let Z be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dimZ=dimX−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≤κ(Z)<dimZ.

Connections between the geometry of a projective variety and of an ample section / Andreatta, Marco; C., Novelli; Occhetta, Gianluca. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - STAMPA. - 2006, 279:13-14(2006), pp. 1387-1395.

Connections between the geometry of a projective variety and of an ample section

Andreatta, Marco;Occhetta, Gianluca
2006

Abstract

Let X be a smooth complex projective variety and let Z be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dimZ=dimX−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≤κ(Z)
13-14
Andreatta, Marco; C., Novelli; Occhetta, Gianluca
Connections between the geometry of a projective variety and of an ample section / Andreatta, Marco; C., Novelli; Occhetta, Gianluca. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - STAMPA. - 2006, 279:13-14(2006), pp. 1387-1395.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/70449
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