On linear systems with multiple points on a rational normal curve

TL;DR

This paper derives a closed formula for the dimension of linear systems in projective space with points on a rational normal curve, explaining their speciality through geometric subvarieties.

Contribution

It provides a new explicit formula for these linear systems and offers a geometric explanation for their speciality based on base locus subvarieties.

Findings

Closed formula for the dimension of linear systems with points on a rational normal curve

Geometric explanation of linear system speciality involving subvarieties

Identification of base locus components affecting system dimension

Abstract

We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.