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

## TL;DR

This paper investigates the structure of $r$-tuples of homogeneous forms in projective space whose common zeros form positive-dimensional sets, revealing that maximal components are either forms vanishing on a line or failing to intersect properly.

## Contribution

It characterizes the components of maximal dimension of the locus of forms with positive-dimensional common zeros, identifying specific geometric configurations.

## Key findings

- Maximal components either vanish on a line or have improper intersections.
- Provides a classification of the geometric structure of these loci.
- Enhances understanding of hypersurface intersections in projective space.

## 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.