# Collections of hypersurfaces containing a curve

## TL;DR

This paper investigates the structure of hypersurface collections in projective space, identifying the dominant components where intersections contain lines, and applies these insights to singular and line-rich hypersurfaces.

## Contribution

It characterizes the largest components of hypersurface collections with positive dimensional intersections, especially those containing lines, and extends results to singular and line-rich hypersurfaces.

## Key findings

- Largest component corresponds to hypersurfaces containing a line
- Identifies dominant components for hypersurfaces with positive dimensional singular loci
- Provides new results on hypersurfaces with more lines through a point than expected

## Abstract

We consider the closed locus of $r$-tuples of hypersurfaces in $\mathbb{P}^r$ with positive dimensional intersection, and show in a large range of degrees that its largest component is the locus of $r$-tuples of hypersurfaces whose intersection contains a line. We then apply our methods to obtain new results on the largest components of the locus of hypersurfaces with positive dimensional singular loci and the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.