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

In this paper, we study minimal generators of the (saturated) defining ideal of ${\sigma }_k(v_d({\mathbb{P}}^n))$ in ${\mathbb{P}}^N$ with $N=\left(\genfrac{}{}{0px}{}{n+d}{d}\right)-1$, the $k$-secant variety of $d$-uple Veronese embedding of projective $n$-space, of a relatively small degree. We first show that the prime ideal $I({\sigma }_4(v_3({\mathbb{P}}^3)))$ can be minimally generated by 36 homogeneous polynomials of degree $5$. It implies that ${\sigma }_4(v_3({\mathbb{P}}^3))\subset {\mathbb{P}}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{d}\mathrm{e}\mathrm{g}({\sigma }_4(v_3({\mathbb{P}}^3)))=105$ and the sectional genus $\pi ({\sigma }_4(v_3({\mathbb{P}}^3)))=316$) and provides a new example of an arithmetically Gorenstein variety of codimension $4$. As an application, we decide non-singularity of a certain locus in ${\sigma }_4(v_3({\mathbb{P}}^3))$. By inheritance, generators of $I({\sigma }_4(v_3({\mathbb{P}}^n)))$ are also obtained for any $n\ge 3$. We also propose a procedure to compute the first non-trivial degree piece $I({\sigma }_k(v_d({\mathbb{P}}^n)){)}_{k+1}$ for a general $k$-th secant case, in terms of prolongation and weight space decomposition, based on the method used for ${\sigma }_4(v_3({\mathbb{P}}^3))$ and treat a few more cases of $k$-secant varieties of the Veronese embedding of a relatively small degree in the end.