Projective normality of finite group quotients and EGZ theorem

TL;DR

This paper proves the projective normality of quotients of finite-dimensional complex vector spaces by finite cyclic groups using toric variety methods and derives the Erdős–Ginzburg–Ziv (EGZ) theorem as a consequence.

Contribution

It establishes projective normality for finite cyclic group quotients of complex vector spaces and connects this geometric property to the EGZ theorem, a combinatorial result.

Findings

Proves projective normality of $P(V)/G$ for finite cyclic groups.

Uses toric variety techniques to establish geometric properties.

Derives the EGZ theorem from the geometric framework.

Abstract

In this note, we prove that for any finite dimensional vector space $V$ over $\mathbb {C}$, and for a finite cyclic group $G$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$ by a method using toric variety, and deduce the EGZ theorem as a consequence.