Let $X\subset \Ps^{2m+1}$ be a projective variety with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a smooth hypersurface $F$ of degree $n>k$. Denote by $NS_m(F)$ and $NS_m(X)$ the $m$-th N\'eron-Severi groups. We prove that if $rkNS_m(F)=1$ then $rkNS_m(X)=1$. Moreover we prove that if $F\in\mid\ic_{X,\Ps^{2m+1}}(n)\mid$ is general and $n>max\{k, 2m+1\}$, then the natural map $NS_m(X)\otimes\bQ \to NS_m(F)\otimes \bQ$ is surjective. When $X$ is a threefold we deduce that $X$ is factorial if and only if $rkNS_2(F)=1$. This allows us to prove the existence of factorial threefolds with many singularities.

Di Gennaro, V., & Franco, D. (2008). Factoriality and Néron-Severi groups. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 10, 745-764.

Factoriality and Néron-Severi groups

DI GENNARO, VINCENZO;
2008

Abstract

Let $X\subset \Ps^{2m+1}$ be a projective variety with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a smooth hypersurface $F$ of degree $n>k$. Denote by $NS_m(F)$ and $NS_m(X)$ the $m$-th N\'eron-Severi groups. We prove that if $rkNS_m(F)=1$ then $rkNS_m(X)=1$. Moreover we prove that if $F\in\mid\ic_{X,\Ps^{2m+1}}(n)\mid$ is general and $n>max\{k, 2m+1\}$, then the natural map $NS_m(X)\otimes\bQ \to NS_m(F)\otimes \bQ$ is surjective. When $X$ is a threefold we deduce that $X$ is factorial if and only if $rkNS_2(F)=1$. This allows us to prove the existence of factorial threefolds with many singularities.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/03 - Geometria
English
Con Impact Factor ISI
Factoriality, N\'eron-Severi group, Noether-Lefschetz Theory, Monodromy, Complete intersection isolated singularity, Milnor fibre.
http://www.worldscinet.com/ccm/10/1005/S021919970800296X.html
Di Gennaro, V., & Franco, D. (2008). Factoriality and Néron-Severi groups. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 10, 745-764.
DI GENNARO, V; Franco, D
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/51891
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