Toric varieties with isolated singularity and smooth normalization

TL;DR

This paper characterizes the generators of the semigroup of certain toric varieties with isolated singularities and smooth normalization, providing explicit descriptions for specific cases.

Contribution

It introduces a prenormal form for semigroup generators and describes the semigroup explicitly for varieties in specific dimensions.

Findings

Complete semigroup description for varieties in ^{2n}

Set of generators for the ideal defining toric surfaces in ^4

Prenormal form for generators of the semigroup

Abstract

In this work, we describe a prenormal form for the generators of the semigroup of a toric variety $X \subset \mathbb{C}^p$ with isolated singularity at the origin and smooth normalization. A complete description of the semigroup is given when $X$ is a variety of dimension $n$ in $\mathbb{C}^{2n}$. Moreover, for toric surfaces in $\mathbb{C}^4$, we provide a set of generators of the ideal $I$ defining $X$.