On the prime ideals of higher secant varieties of Veronese embeddings of small degrees

Abstract

In this paper, we study minimal generators of the (saturated) defining ideal of the $k$-secant variety ${\sigma }_k(v_d({\mathbb{P}}^n))$ of the image of the $d$-uple Veronese embedding $v_d:{\mathbb{P}}^n\to {\mathbb{P}}^N$ with $N=\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-1$, focusing on cases where the degree of ${\sigma }_k(v_d({\mathbb{P}}^n))$ is relatively small. First, we show that the prime ideal $I({\sigma }_4(v_3({\mathbb{P}}^3)))$ is minimally generated by $36$ homogeneous polynomials of degree $5$. This implies that ${\sigma }_4(v_3({\mathbb{P}}^3))\subset {\mathbb{P}}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}({\sigma }_4(v_3({\mathbb{P}}^3)))=105$ and the sectional genus $\pi ({\sigma }_4(v_3({\mathbb{P}}^3)))=316$), thereby providing a new example of an arithmetically Gorenstein variety of codimension $4$. This result addresses the symmetric version of the ``Salmon problem'' posed by E. Allman in \cite{Allman}. As an application, we decide the non-singularity of a certain locus in ${\sigma }_4(v_3({\mathbb{P}}^3))$. Furthermore, by inheritance, we obtain the generators of $I({\sigma }_4(v_3({\mathbb{P}}^n)))$ for all $n\ge 3$. Based on the method used for ${\sigma }_4(v_3({\mathbb{P}}^3))$, we also propose a procedure to compute the first non-trivial degree piece, $I({\sigma }_k(v_d({\mathbb{P}}^n)){)}_{k+1}$, for the general $k$-secant case using prolongation and weight space decomposition. Applying this procedure, we present a few more cases of $k$-secant varieties of relatively small degrees; in each of these cases, the ideal is generated in degree $k+1$ and can be fully determined by explicitly computing all generators within this degree piece.

27 pages, Some remarks and Macaulay2 implementation added, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze