Hypersurfaces cutting out a projective variety

Abstract

Let X X be a nondegenerate projective variety of degree d d and codimension e e in a projective space P N \mathbb {P}^{N} defined over an algebraically closed field. We study the following two problems: Is the length of the intersection of X X and a line L L in P N \mathbb {P}^{N} at most d − e + 1 d-e+1 if L ⊈ X L \not \subseteq X ? Is the scheme-theoretic intersection of all hypersurfaces of degree at most d − e + 1 d-e+1 containing X X equal to X X ? To study the second problem, we look at the locus of points from which X X is projected nonbirationally.