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 $q = p^r$.



Effective cone of $C \times C$ where $C$ is a Fermat curve
Since I asked this question several months ago, I wrote a paper proving that the Mori cone of $C \times C$ for a smooth projective complex curve $C$ of genus $g \geq 2$ is not polyhedral. I thought I'd include a link to that for completeness.


<http://arxiv.org/abs/1502.06061>


