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

Abstract

Let nu _{d,n}: mathbb {P}^nrightarrow mathbb {P}^r, r=left( {begin{array}{c}n+d nend{array}}right) , be the order d Veronese embedding. For any d_nge 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 Ssubset mathbb {P}^n is a finite set which is the complete intersection of hypersurfaces of degree d_1, dots ,d_n. For any qin check{eta }(n,d;d_1,ldots ,d_n), we prove the uniqueness of the set nu _{d,n}(S) if dge 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 dge 2d_1.