Let $X$ be a projective non-singular curve over an algebraically closed field $k$, as a scheme. I am trying to do exercise 5.3c of Hartshorne chapter III. There I am asked to show that the arithmetic genus of such a curve is a birational invariant. I've shown that the arithmetic genus is given by the dimension of the first cohomology, $\dim_{k}H^{1}(X, \mathcal{O}_{X})$. The problem is I really have no idea how to start. I originally considered using the exact sequence, $$ 0 \longrightarrow j_{!}j^{*} \mathcal{O}_{X} \longrightarrow \mathcal{O}_{X} \longrightarrow i_{*}i^{*} \mathcal{O}_{X} \longrightarrow 0 $$ where $j: U \hookrightarrow X$ is the inclusion of a dense open subset and $i: Z \hookrightarrow X$ is the inclusion of its complement. I was going to take the long exact sequence for cohomology and use the fact that two such curves are birational if they have an isomorphic dense open subset. Although this argument seemed to go nowhere useful.
Is anyone able to at least point me in the right direction?