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

## TL;DR

This paper explicitly describes a minimal binomial generating set for the lattice ideals of certain Veronese 3-fold projections, providing a basis for the associated lattice and identifying minimal generators.

## Contribution

It introduces a method to explicitly determine a minimal binomial generating set for the lattice ideals of specific Veronese 3-fold projections, advancing understanding of their algebraic structure.

## Key findings

- Provides a $ ext{Z}$-basis for the lattice $L_{\eta}$.
- Identifies a minimal binomial generating set for $I(X_d)$.
- Clarifies the algebraic structure of the ideals for these toric varieties.

## 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 $d\ge 4$ 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 $T_d$ are all monomials of degree $d$ in $k[x,y,z,t]$ invariant under the action of the diagonal matrix $M(1,e,e^2,e^3).$ 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 $I(X_d)=I_+(\eta)$.