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.



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> **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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Image of $H^0(C,\omega_C-x)$ in $G(g-1,H^0(C,\omega_C))$
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 $C$.


