A note on projective normality

Abstract:

Let $G$ be any finite group, $G\to GL(V)$ be a representation of $G$, where $V$ is a finite-dimensional vector space over an algebraically closed field $k$. Theorem. Assume that either $char k = 0$ or $char k = p > 0$ with $p \nmid |G|$. Then the quotient variety $\mathbb {P}(V)/G$ is projectively normal with respect to the line bundle $\mathcal {L}$, where $\mathcal {L}$ is the descent of $\mathcal {O}(1)^{\otimes m}$ from $\mathbb {P}(V)$ with $m=|G|!$. This partially solves a question raised in the paper of Kannan, Pattanayak and Sardar [Proc. Amer. Math. Soc. 137 (2009), 863–867].