Abstract Classical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most ( c + 1 2 ) {{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least ( c + 1 2 ) - 3 {{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree ≥ 2 ⁢ c + 1 {\geq 2c+1} .

中文翻译：

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关于包含射影变种的超二次曲面

摘要 经典 Castelnuovo 引理表明，一个非退化不可约射影变数 c 的线性无关二次方程的个数最多为 ( c + 1 2 ) {{{c+1}\choose{2}}} 且等式成立当且仅当多样性是最小程度的。G. Fano 对 Castelnuovo Lemma 的概括也暗示了下一种情况发生当且仅当该变体是 del Pezzo 变体。最近，这些结果扩展到 [E. Park，在包含射影变体的超曲面上，论坛数学。27 2015, 2, 843–875]。本文旨在完成至少满足 ( c + 1 2 ) - 3 {{{c+1}\choose{2}}-3} 线性无关二次方程的品种分类。