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

Abstract

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 $d \geq 4$ and any $d$-th root $e$ of 1 we denote by $X_d$ the toric variety defined as the image of the morphism $\varphi_{T_d}: \mathbb{P}^3 \longrightarrow \mathbb{P}^{\mu\left(T_d\right)-1}$ where $T_d$ are all monomials of degree $d$ in $k[x, y, z, t]$ invariant under the action of the diagonal matrix $M\left(1, e, e^2, e^3\right)$. In this work, we describe a $\mathbb{Z}$-basis of the lattice $L_\eta$ associated to $I\left(X_d\right)$ as well as a minimal binomial set of generators of the lattice ideal $I\left(X_d\right)=I_{+}(\eta)$.