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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 $C$?

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First, note that projectification provides a natural isomorphism $$G(g-1, \, H^0(C, \omega_C)) \simeq \mathbb{G}(g-2, \, \mathbb{P}H^0(C, \omega_C))=\mathbb{P}H^0(C, \omega_C)^*.$$

Next, observe that the canonical map $$\varphi \colon C \to \mathbb{P}H^0(C, \omega_C)^*$$ associates to $x \in C$ the hyperplane $\mathbb{P}H^0(C, \, \omega_C - x) \subset \mathbb{P}H^0(C, \omega_C)$.

Thus, the locus you are interested in is just the canonical image of $C$.

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