LAUSR

Let $$\nu _{d,n}: \mathbb {P}^n\rightarrow \mathbb {P}^r$$νd,n:Pn→Pr, $$r=\left( {\begin{array}{c}n+d\\ n\end{array}}\right) $$r=n+dn, be the order d Veronese embedding. For any $$d_n\ge \cdots \ge d_1>0$$dn≥⋯≥d1>0 let $$\check{\eta }(n,d;d_1,\ldots ,d_n)\subseteq \mathbb {P}^r$$ηˇ(n,d;d1,…,dn)⊆Pr be the union of all linear spans of $$\nu _{d,n}(S)$$νd,n(S) where $$S\subset \mathbb {P}^n$$S⊂Pn is a finite set which is the complete intersection of hypersurfaces of degree $$d_1, \dots ,d_n$$d1,⋯,dn. For any $$q\in \check{\eta }(n,d;d_1,\ldots ,d_n)$$q∈ηˇ(n,d;d1,…,dn), we prove the uniqueness of the set $$\nu _{d,n}(S)$$νd,n(S) if $$d\ge d_1+\cdots +d_{n-1}+2d_n-n$$d≥d1+⋯+dn-1+2dn-n and q is not spanned by a proper subset of $$\nu _{d,n}(S)$$νd,n(S). We compute $$\dim \check{\eta }(2,d;d_1,d_1)$$dimηˇ(2,d;d1,d1) when $$d\ge 2d_1$$d≥2d1.