In this article we prove, for canonical model of curves, a theorem illustrating the general principle that (to paraphrase Arnold) any homogeneous ring that has a serious reason for being quadratically presented is Koszul. In this case we give a new proof, which is both elementary and geometric, of a theorem of Finkelberg and Vishik [VF] (see also [Po]) which says that whenever the canonical ring of a smooth complex projective curve is quadratically presented, it is Koszul. Our method is different from [Po]. We use vector bundle technique, building upon the one used in [GL]. We would also like to mention here that our methods fit a more general principle as shown in [GP1], [GP2] and [GP3].
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