Canonical ring of a curve is Koszul: A simple proof (Q1358734)

Let \(k\) be a field. A (commutative) graded \(k\)-algebra of the form \(R:=k\oplus R_1\oplus\cdots\oplus R_n\dots\) is said to be Koszul if its Koszul complex is exact, or, equivalently, if \(k=R/R_{>0}\) has a linear minimal resolution over \(R\); namely \(\cdots\to E_p\to E_{p-1}\to \cdots\to E_2\to E_1\to E_0\to k\to 0\) with \(E_0=R\) and \(E_p=R(-p)^{\oplus r(p)}\) for any \(p\geq 1\). In this article we give a new proof, which is both elementary and geometric, of a theorem by \textit{A. Vishik} and \textit{M. Finkelberg} [J. Algebra 162, No. 2, 535-539 (1993; Zbl 0819.14008); see also \textit{A. Polishchuk}, J. Algebra 178, No. 1, 122-135 (1995; Zbl 0861.14030)] which says that whenever the canonical ring of a smooth complex projective curve is quadratically presented, it is Koszul. Our method is different from Polishchuk's (loc. cit.). We use vector bundle technique, building upon the one used by \textit{M. Green} and \textit{R. Lazarsfeld} [in: Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 129-142 (1985; Zbl 0577.14018)].