Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 109–113 | Cite as

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

  • Edoardo BallicoEmail author
Open Access


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 \(d\ge 2d_1\).

Mathematics Subject Classification

14N05 15A69 



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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly

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