Global sections of canonical sheaf and genus

Question

Let $k$ be algebraically closed and C a curve. Then $g(C) = h^0(C, \omega_C)$. I want to show that if all the degree $0$ line bundles are trivial, then $g = 0$. I can sort of see that if I think of $C$ as a compact complex surface, then being able to shrink down every degree $1$ loop on the surface to a point implies that the surface has no holes. But I am unsure how to obtain a proper geometric argument. Thanks.

Answer 1

Let $C$ be a complete smooth curve $C$ of genus $g$ over an algebraically closed field.

An explicit approach
Fix a point $p_0\in C$. Then, if $g\gt 0$, the line bundles $\mathcal O(1.p-1.p_0) \: (p\in C)$ associated to the divisors of degree zero $1.p-1.p_0\in \operatorname {Div}_0(C)$ are pairwise non isomorphic.

A conceptual approach
The curve $C$ has a Jacobian, which is an abelian variety of dimension $g$ and whose closed points parametrize the isomorphism classes of line bundles of degree zero on $C$ .
So indeed it is only for $g=0$ that all line bundles on $C$ are trivial.