Collections of Hypersurfaces Containing a Curve

Abstract

Abstract We consider the closed locus parameterizing $k$-tuples of hypersurfaces that have positive dimensional intersection and fail to intersect properly, and show in a large range of degrees that its unique irreducible component of maximal dimension consists of tuples of hypersurfaces whose intersection contains a line. We then apply our methods in conjunction with a known reduction to positive characteristic argument to find the unique component of maximal dimension of the locus of hypersurfaces with positive dimensional singular loci. We will also find the components of maximal dimension of the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.