# What is Koszul dual of a curve?

Let $X$ be a curve embedded into a projective space $\mathbb P$ such that it is cut out (scheme-theoretically or ideal-theoretically) by quadrics.

What is known about the Koszul dual of the homogeneous coordinate ring of $X$? What is the center of the Koszul dual ring?

Edited: I would be thankful for references regarding even the very basic special cases. For example, $X$ a rational normal curve; or $X$ is an elliptic curve embedded by a full linear system of degree at least $4$. Or, $X$ is any curve, and the divisor is sufficiently large.

• Being cut out by quadrics isn't sufficient to make the homogeneous coordinate ring koszul. – Phil Tosteson Jan 23 '16 at 21:48
• Of course. There is still a concept of Koszul dual, though. – Lev Borisov Jan 23 '16 at 21:52
• Great question! This paper of Bezrukavnikov arxiv.org/abs/alg-geom/9502021 (apparently never published?) is relevant for $\mathbb{P}^1$. – Geordie Williamson Feb 9 '16 at 15:33
• See also: Inamdar, Mehta, Frobenius splitting of Schubert varieties and linear syzygies. American Journal of Math. – Geordie Williamson Feb 9 '16 at 15:39