Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.

Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.

DI GENNARO, V., Franco, D., & Marini, G. (2016). Algebraic versus homological equivalence for singular varieties. COMMUNICATIONS IN ALGEBRA, 44(6), 2547-2560 [10.1080/00927872.2015.1053904].

Algebraic versus homological equivalence for singular varieties

Di Gennaro Vincenzo;Marini Giambattista
2016

Abstract

Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03 - Geometria
English
Let Y ⊆ ℙN be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimYsing ≤ min {d + h − 1, n − 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H2(d+h)(Y; ℚ) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.
Algebraic cycle; Algebraic equivalence; Chow variety; Connectivity Theorem; Hilbert scheme; Homological equivalence; Projective variety; Singularity;
DI GENNARO, V., Franco, D., & Marini, G. (2016). Algebraic versus homological equivalence for singular varieties. COMMUNICATIONS IN ALGEBRA, 44(6), 2547-2560 [10.1080/00927872.2015.1053904].
DI GENNARO, V; Franco, D; Marini, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/198043
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