Koszul algebras associated to adjunction bundles
Recall that a graded algebra \(R = \bigoplus_{s \geq 0}R_ s\) over a field \(k = P_ 0\) is called ``Koszul'' iff all the graded modules \(\text{Tor}^ R_ m (k,k)\) are concentrated in degree \(m\). Examples of Koszul rings are: regular rings (the Koszul complex!), Segre-Veronese rings [cf. \textit{Ş. Barcanescu} and the reviewer, C. R. Acad. Sci., Paris, Sér. A 288, 237-239 (1979; Zbl 0402.13019)], the rings \(R(L^ s):=\bigoplus_{m\geq 0}H^ 0(X,L^{sm})\) associated, for \(s\) big enough, to a very ample line bundle \(L\) on a scheme \(X\) over an algebraically closed field \(k\) [cf. \textit{J. Backelin} in Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 79-100 (1986; Zbl 0588.13011)]. In this paper one proves that \(R(K_ X \otimes L^ d)\) is Koszul, for \(X\) a smooth projective variety over the complex field \(\mathbb{C}\), \(L\) a very ample line bundle on \(X\) and \(d \geq \dim X+2\). This result is also extended to \(d = \dim X+1\).
scientific article; zbMATH DE number 202988