A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of genus $\geq 2$ for which we understand the effective cone of $C \times C$ and its image in $\text{Num}(C \times C)$? In particular, when can we say that the image in $\text{Num}(C \times C)$ is finitely generated?

Edit: Since there haven't been any answers to my question about finite generation, perhaps I can be more specific. Let $k = \mathbb{C}$. If $C = \mathbb{V}(x^n + y^n + z^n) \subset \mathbb{P}^2_k$, are there any techniques to calculate the effective cone of $C \times C$ when $n \geq 4$?

I ask this because if $k$ is a field of characteristic $p > 0$, it's well-known that this cone is not finitely generated since we have infinitely many extremal curves $\Gamma_q := \{(u,v) | \text{ Frob}_q(u) = v \}$ where $q = p^r$.