Complete families of immersed curves

TL;DR

This paper extends the understanding of families of immersed curves in projective space and generalized flag varieties, providing sharp bounds and identifying key positivity conditions.

Contribution

It establishes a sharp bound for the dimension of complete families of smooth rational immersed curves in projective space and identifies positivity conditions that limit such families in flag varieties.

Findings

Sharp bound for rational immersed curves in projective space

Positivity condition on tangent bundle restricts immersed curves in flag varieties

Complete families of positive genus immersed curves are limited in generalized flag varieties

Abstract

We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space, completing the picture in projective space. In another direction, we isolate the necessary positivity condition on the tangent bundle of projective space used to run the argument, which allows us to rule out large dimensional families of immersed curves of positive genus in generalized flag varieties.