# ag.algebraic geometry – Curve with no embedding in a toric surface

I am looking for a smooth proper curve $$C$$ such that there does not exist any closed embedding $$C to S$$ where $$S$$ is a (normal projective) toric surface.

Since $$C$$ is smooth I believe it suffices to consider smooth projective toric surfaces $$S$$ since we may always perform a toric resolution of singularities and the strict transform of $$C$$ will be isomorphic to $$C$$ since $$C$$ is smooth.

Using the result on p.25 of Harris Mumford, On the kodaira dimension of the moduli space of curves, I can conclude that a very general curve cannot have any such embedding.

However, I am not able to write down an explicit example. Does anyone know such an example or what sort of obstruction might work to check this in particular examples.