Minimal set of binomial generators for certain Veronese 3-fold projections

The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any $d$-th root $e$ of 1 we denote by $X_d$ the toric variety defined as the image of the morphism $\varphi _{T_d}:$\mathbb{P}$^3 \longrightarrow $\mathbb{P}$^{\mu (T_d)-1}$where$T_d$areallmonomialsofdegree$d$in$k[x,y,z,t]$ invariant under the action of the diagonal matrix $M(1,e,e^2,e^3)\mathrm{.}$ In this work, we describe a $\mathbb{Z}$-basis of the lattice $L_{\eta }$ associated to $I(X_d)$ as well as a minimal binomial set of generators of the lattice ideal $I(X_d)=I_{+}(\eta )$.