On hypersurfaces containing projective varieties

Abstract

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 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.

Sums of Squares and Quadratic Persistence on Real Projective Varieties

“…Proposition 5.10 in [Zak99] proves that a projective subvariety X ⊂ P n with ε(X) = 1 is a hypersurface of degree at least 3 or linearly-normal variety such that deg(X) = 2 + codim(X). Corollary 1.4 in [Par15] proves that, for a subvariety X ⊂ P n satisfying codim(X) 3 and ε(X) = 2, the pair deg(X), depth(X) is either 2 + codim(X), dim(X) or 3 + codim(X), 1 + dim(X) .…”

Collections of Hypersurfaces Containing a Curve

“…Note that Lemma 1.10 is also elementary, as it can be proven by repeatedly cutting it with a general hyperplane and applying [10,Lemma 3.1]. The varieties X for which equality is satisfied are the varieties of minimal degree [14,Remark 1.6 (1)].…”

“…Let Span(b) ⊂ Hilb r denote the integral curves whose span is exactly a b-dimensional plane except for the rational normal curves of degree b.From a special case of[14, Theorem 4.5], we have Lemma 6.16. We haveh Span(b) ∩ Hilb 1 P r (d) ≥ (b + 1)d.…”