For a smooth toric projective variety $X$, I'm able to use the result
$H^0(X,\mathcal{O}(D)) \cong \bigoplus_{m \in \Delta_D \cap M} \mathbb{C}\cdot \chi^m$
to understand global sections of a given sheaf on $X$, subject to the choice of a torus-invariant divisor $D = \sum_{\rho\in \Sigma(1)} a_\rho D_\rho$, based on the corresponding polytope $\Delta_D = \{m\in M\text{ }|\text{ }\left<m,u_\rho\right>\geq -a_i\}$ for $u_\rho$ a primitive ray generator of $\rho\in \Sigma(1)$.
This helps me understand $H^0(\mathbb{P}^n, \mathcal{O}(D))$ for instance, for a given $D$.
If we identify $\mathbb{P}^1_\mathbb{C}$ with a genus 0 Riemann surface, then I can understand global sections of a genus 0 Riemann surface via this correspondence with a toric variety. I am wondering: Are any positive genus Riemann surfaces toric, so that I can understand the global sections of a given sheaf in the same way? If they are not toric, why are they not toric?