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?