Zero-dimensional complete intersections and their linear span in the Veronese embeddings of projective spaces

Abstract

Let $\nu _{d,n}: \mathbb {P}^n\rightarrow \mathbb {P}^r$, $r=\left( {\begin{array}{c}n+d\\ n\end{array}}\right)$, be the order d Veronese embedding. For any $d_n\ge \cdots \ge d_1>0$ let $\check{\eta }(n,d;d_1,\ldots ,d_n)\subseteq \mathbb {P}^r$ be the union of all linear spans of $\nu _{d,n}(S)$ where $S\subset \mathbb {P}^n$ is a finite set which is the complete intersection of hypersurfaces of degree $d_1, \dots ,d_n$. For any $q\in \check{\eta }(n,d;d_1,\ldots ,d_n)$, we prove the uniqueness of the set $\nu _{d,n}(S)$ if $d\ge d_1+\cdots +d_{n-1}+2d_n-n$ and q is not spanned by a proper subset of $\nu _{d,n}(S)$. We compute $\dim \check{\eta }(2,d;d_1,d_1)$ when $d\ge 2d_1$.