  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.
