Complete families of linearly non-degenerate rational curves

TL;DR

The paper establishes upper bounds on the moduli of complete families of linearly non-degenerate rational curves in projective space, depending on their degree, using cohomological analysis of associated maps.

Contribution

It provides new bounds on the moduli of such families for degrees greater than 2 and for degree 2, employing a novel cohomological approach.

Findings

For degree e > 2, at most n-1 moduli.

For degree e = 2, at most n moduli.

Method involves analyzing maps to Grassmannians and cohomology.

Abstract

We prove that a complete family of linearly non-degenerate rational curves of degree $e > 2$ in $\mathbb{P}^n$ has at most $n-1$ moduli. For $e = 2$ we prove that such a family has at most $n$ moduli. It is unknown whether or not this is the best possible result. The general method involves exhibiting a map from the base of a family $X$ to the Grassmaninian of $e$-planes in $\mathbb{P}^n$ and analyzing the resulting map on cohomology.