Projective normality of quotient varieties modulo finite groups

TL;DR

This paper proves that quotient varieties of finite-dimensional vector spaces by certain finite groups are projectively normal under specific conditions, extending understanding of their geometric properties.

Contribution

It establishes projective normality of quotient varieties for groups generated by pseudo reflections or solvable groups, under conditions on the field and group order.

Findings

Quotient varieties are projectively normal under the given conditions.

The result applies to groups generated by pseudo reflections and solvable groups.

The proof relies on properties of descent of line bundles and group actions.

Abstract

In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.