Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point.

Question:What is the image of $H^0(C,\omega_C-x)$ in the Grassmannian $G(g-1,\, H^0(C,\omega_C))$ as $x$ varies along $C$?