The coordinate ring of general curve of genus \(g\geq 5\) is Koszul (Q1320218)

The graded algebra \(A^ \bullet\) over \(\mathbb{C}\) is called a Koszul algebra if \(A^ 0 = \mathbb{C}\), \(A^ \bullet\) is generated by \(A^ 1\), \(\dim A^ 1 < \infty\), whereas the ideal of relations is generated by quadratic relations \(I^{(2)} \subset (A^ 1)^{\otimes 2}\), and the following condition is fulfilled: \(\text{Ext}^ \bullet_ A (\mathbb{C}, \mathbb{C})\) is generated by \(\text{Ext}^ 1_ A (\mathbb{C}, \mathbb{C})\). In this case \(\text{Ext}^ \bullet_ A (\mathbb{C}, \mathbb{C}) = A'\) as graded algebras, where \(A'\) denotes the algebra dual to \(A^ \bullet\), -- Koszul algebras are very interesting objects from the point of view of the homological algebra (Bernstein-Gelfand-Gelfand duality). Remarkable examples of Koszul algebras are provided by noncommutative algebraic geometry (quantum groups). -- In this note we prove that the canonical coordinate ring of a general smooth curve of genus \(g \geq 5\) is Koszul. More precisely, the following theorem holds: Main theorem. If \(C\) is a nonhyperelliptic, nontrigonal, nonsuperelliptic curve of genus \(g \geq 5\), and \(C\) is not a plane quintic, then the projective coordinate ring of \(C\) in the canonical embedding \(R^ \bullet : = \bigoplus_{n \geq 0} H^ 0 (C,K^ n)\) (where \(K\) is the canonical bundle) is quadratic Koszul.