# Ravi Vakil's book on Algebraic Geometry , Exercice 19.2.A

In what follows, $$C$$ will be a projective, geometrically regular, geometrically integral curve over a field $$k$$, and $$\mathcal{L}$$ is an invertible sheaf on $$C$$.

$$19.2.A.$$ EXERCISE. Suppose $$\mathcal{L}$$ is a degree $$2g - 2$$ invertible sheaf. Show that it has $$g - 1$$ or $$g$$ sections, and it has $$g$$ sections if and only if $$\mathcal{L}\cong \omega_C$$.

What does it mean $$\mathcal{L}$$ to have $$g-1$$ or $$g$$ sections?

Is it $$h^{0}(C, \mathcal{L})= g-1$$ or $$g$$? Would it be this?

Because by applying Riemann-Roch directly, we get

$$h^0(C, \mathcal{L}) - h^1(C, \mathcal{L})= \text{deg} \mathcal{L} -g+1= g-1$$.

But what about $$h^1(C, \mathcal{L})$$?

• You are correct that you are trying to show $h^0(C,\mathcal L) = g-1$ or $g$. Riemann-Roch reduces you to showing $h^1(C,\mathcal L) = 0$ or $1$. This is what Serre Duality is for: it tells us that $h^1(C,\mathcal L) = h^0(C,\omega_C\otimes\mathcal L^\vee)$, where $\vee$ denotes the dual line bundle. Notice that $\deg (\omega_C\otimes\mathcal L^\vee) = 0$. Now, you should have proved a lemma at some point by now that a degree $0$ line bundle has a unique section if and only if it is isomorphic to $\mathcal O_C$, which in this case is equivalent to $\mathcal L \cong \omega_C$. – Tabes Bridges 2 days ago