Frobenius splittings and invariant theory
In the article a criterion is given for quotients of linear representations in positive characteristic to be Cohen-Macaulay (C-M). More generally, the authors prove the following: Let \(G \to \text{ SL}(W)\) be a representation of a reductive group \(G\) over some algebraically closed field \(k\) of positive characteristic and assume that the associated GIT-quotient \(\pi : {\mathbb{P}}(W)^{ss} \to Q\) satisfies the following conditions: (1) \(Q\) is Frobenius-split, i.e., for the absolute Frobenius morphism \(F : Q \to Q\) the sequence \(0 \to {\mathcal O}_{Q} \to F_{*} {\mathcal O}_{Q}\) splits, (2) each sheaf \((\pi_{*} {\mathcal O}(j))^{G}\) is C-M, (3) outside a saturated subset of codimension at least two in \(\mathbb{P}(W)^{ss}\) the map \(\pi\) is a principal \(G\)-bundle. Then \({\mathcal O}(W)^{G}\) is \(F\)-regular [in the sense of \textit{M. Hochster} and \textit{C. Huneke}, Bull. Am. Math. Soc., New Ser. 18, No. 1, 45-48 (1988; Zbl 0674.13003)] and thus \(F\)-pure and C-M. Moreover, if \(\chi\) is a character of finite order of \(G\) such that all sheaves \((\pi_{*}({\mathcal O}(j) \otimes \chi))^{G}\) are C-M, then the \({\mathcal O}(W)^{G}\)-module of covariants with respect to \(\chi\) is C-M. The above result is applied to the following situation: Let \(G := \text{ SL}(3,k)\) act via adjoint representation on the vector space \(M_{3}\) of \(3 \times 3\)-matrices over \(k\) and assume that the characteristic of \(k\) is at least 4. The authors show that the quotient \(M^{g} / / G\) is C-M if \(g \geq 3\). Moreover, it is concluded that the moduli space of semistable vector bundles of rank 3 and degree 0 on a smooth projective curve of genus \(g > 3\) is C-M.
algebraic quotients of linear representations
moduli space of semistable vector bundles
scientific article; zbMATH DE number 1112697