Incident rational curves

TL;DR

This paper investigates families of rational curves on algebraic varieties with incidence conditions, proving an analogue of bend-and-break and establishing irreducibility in specific cases.

Contribution

It introduces a bend-and-break analogue for rational curves satisfying incidence conditions and proves irreducibility for certain families in projective space.

Findings

Families must contain reducible curves under certain conditions

In specific cases, the family of rational curves is irreducible

Provides new tools for studying rational curves on algebraic varieties

Abstract

We study families of rational curves on an algebraic variety satisfying incidence conditions. We prove an analogue of bend-and-break: that is, we show that under suitable conditions, such a family must contain reducibles. In the case of curves in $\P^n$ incident to certain complete intersections, we prove the family is irreducible.