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 the $k$-secant variety $σ_k(v_d(\mathbb{P}^n))$ of the image of the $d$-uple Veronese embedding $v_d: \mathbb{P}^n \rightarrow \mathbb{P}^N$ with ${N=\binom{n+d}{d}-1}$, focusing on cases where the degree of $σ_k(v_d(\ma