On hypersurfaces containing projective varieties

TL;DR

This paper extends classical results on the equations defining projective varieties, generalizing Castelnuovo's Lemma and related theorems to higher dimensions and more complex cases.

Contribution

It provides new generalizations of classical lemmas, characterizing projective varieties by their defining equations in arbitrary dimensions.

Findings

Characterization of varieties of minimal degree

Extension of Fano's generalization to higher dimensions

Identification of cases for varieties with specific equation counts

Abstract

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.