On hypersurfaces containing projective varieties

Details On hypersurfaces containing projective varieties On hypersurfaces containing projective varieties

2011-04-29

arxiv.org/abs/1104.5599v1

Mathematics

Algebraic Geometry

28 pages, 1 figure

Scientific paper

Classical Castelnuovo's Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension $c$ is at most ${{c+1} \choose {2}}$ and the equality is attained if and only if the variety is of minimal degree. Also a generalization of Castelnuovo's Lemma by G. Fano implies that the next case occurs if and only if the variety is a del Pezzo variety. For curve case, these results are extended to equations of arbitrary degree respectively by J. Harris and S. L'vovsky. This paper is intended to extend these results to arbitrary dimensional varieties and to the next cases.

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