Let $k$ be a field. Let $g\geq 0$ be an integer.
I have an elementary question.
Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of genus $g$. (Note that $N$ can also be $\infty$.)
Is $N$ finite if $k$ is finite?
When is $N$ finite in general?
I'm looking for the "most elementary" answer to this question.