A note on projective normality

Abstract

Abstract. Let G be any finite group, G → 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 |G|. Then the quotient variety P(V )/G is projectively normal with respect to the line bundle L, where L is the descent of O(1) ⊗m from P(V ) with m = |G|!. This partially solves a question raised in the paper of Kannan, Pattanayak and Sardar [Proc.

Projective normality of torus quotients of flag varieties

“…Since E 1,3 = 5, we have N 4,2 = k and N 3,2 = k. So we have E 2k,2 = 4. Hence we conclude that, Row 2k = (2,4,6). Then p 135 p 246 ∈ R 1 and divides M. So by induction we are done.…”

“…They prove that the descent of O X (1) |G| is projectively normal. In [4], these results were obtained for every finite group but with a larger power of the descent of O X (1) |G| . In [12], there was an attempt to study projective normality of T \\(G 2,n ) (n odd) with respect to the descent of the line bundle corresponding to the fundamental weight ω 2 .…”

“…Similarly E 1,2 cannot be 5. So, Row 1 can be one of the elements from the set { (1,2,3), (1,2,4), (1,2,5), (1,3,4), (1,3,5)}.…”

“…Since E 1,2 = 2 we have N 3,1 ≥ 1 and so E 2k,1 = 3. So Row 2k = (3,4,6). Then p 125 p 346 ∈ R 1 and is a factor of M.…”

“…We are now left with two cases, either Row 1 = (1,2,3) or Row 1 = (1,2,4) Case -1 Row 1 = (1,2,4) Since E 1,3 = 4 we have N 5,3 < k. Since N 5,1 = 0 it follows that N 5,2 ≥ 1 and hence, E 2k,2 = 5. If N 4,1 = 0 then N 3,1 ≥ 1.…”