Nef cone and successive minima: an example

In this paper, we compute the nef cone and the pseudo-effective cone of $C\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\times J$ has the minimal Picard number. As a consequence, we also compute the successive minima of a height function for the relative setting $C\times J\to J$, and our result shows that Zhang’s theorem of successive minima does not hold in this case.