Interpolation for degree 2 Veroneses of odd dimension

TL;DR

The paper proves that for odd dimensions, a large number of degree 2 Veroneses pass through a specific set of general points, extending classical interpolation results and addressing open questions.

Contribution

It generalizes classical interpolation results to degree 2 Veroneses in odd dimensions, providing a lower bound on the number of such Veroneses passing through given points.

Findings

Existence of multiple degree 2 Veroneses through general points in odd dimensions

At least 2^{n(n-1)} such Veroneses exist for specified point configurations

Progress on a question by Landesman and Patel, extending Coble's work

Abstract

A classical fact is that through any $d+3$ general points in $\mathbb{P}_\mathbb{C}^d$ there exists a unique rational normal curve of degree $d$ passing through them. We generalize this by proving the following: when $n$ is odd, for any $\binom{n+2}{2} + n+1$ general points in $\mathbb{P}_\mathbb{C}^{\binom{n+2}{2} - 1}$, there exist at least $2^{n(n-1)}$ degree 2 Veroneses passing through them. This makes substantial progress on a question of Aaron Landesman and Anand Patel, and extends the work of Arthur Coble.