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.
Categories: math.AG