A sufficient condition for a projective variety to be the Proj of a Gorenstein graded ring (Q1209595)

Let \(X\) be a normal projective variety over an algebraically closed field \(k\), and \(D\) an ample \(\mathbb{Q}\)-divisor on \(X\), i.e., a rational coefficient Weil divisor whose multiple \(rD\) for some \(r \in \mathbb{N}\) is an ample Cartier divisor. We consider a normal graded ring \(R(X,D)\) defined by \(R(X,D) = \bigoplus^{+ \infty}_{n=0} H^ 0 (X,{\mathcal O}_ X (nD))T^ n\), where \({\mathcal O}_ X (nD)\) is the divisorial sheaf associated with a \(\mathbb{Q}\)-divisor \(nD\). We are interested in finding a criterion for a normal projective variety \(X\) over \(k\) to have an ample \(\mathbb{Q}\)-divisor \(D\) with \(R(X,D)\) Gorenstein. Since \(X = \text{Proj} R(X,D)\), thanks to a theorem of \textit{M. Demazure} [Trav. Cours 37, 35-68 (1988; Zbl 0686.14005)], it is equivalent to ask when a normal projective variety over \(k\) is the Proj of a Gorenstein normal graded \(k\)-algebra. When \(D\) is an ample \(\mathbb{Q}\)-divisor, \textit{K. Watanabe} [Nagoya Math. J. 83, 203-211 (1981; Zbl 0518.13003)] has established a criterion for \(R(X,D)\) to be Gorenstein, in terms of \(D\) and the canonical divisor \(K_ X\) of \(X\). The purpose here is to solve our problem, at least when \(X\) is Gorenstein, based on the criterion of Watanabe (loc. cit). -- Main result: Theorem. Let \(X\) be a Gorenstein normal projective variety of dimension \(N\) over an algebraically closed field \(k\). (a) Suppose that \(H^ i (X,{\mathcal O}_ X) = 0\), \(0<i<N\). Then, for every positive odd integer \(a\), there exists an ample \(\mathbb{Q}\)-divisor \(D\) on \(X\) such that \(R(X,D)\) is a Gorenstein graded ring with \(a\)-invariant \(a(R(X,D))=a\). In particular, \(X\) is the Proj of a Gorenstein normal graded \(k\)-algebra. (b) Suppose furthermore that there exists a Cartier divisor \(F\), with \(H^ i (X,{\mathcal O}_ X (F)) = 0\) for \(0<i<N\), such that \(2F\) is linearly equivalent to a canonical divisor \(K_ X\). Then, for every positive even integer \(a\), there exists an ample \(\mathbb{Q}\)-divisor \(D\) on \(X\) such that \(R(X,D)\) is a Gorenstein graded ring with \(a(R(X,D))=a\).