A note on $r$ hypersurfaces intersecting in $\mathbb{P}^r$

Abstract

We consider the locus of $r$-tuples of homogeneous forms of some fixed degree whose common vanishing locus in $\mathbb{P}^r$ is positive dimensional. We show that any component of maximal dimension of that locus either consists of homogeneous forms all vanishing on some line or homogeneous forms where a proper subset fail to intersect properly.