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.