Let $K$ be a field equipped with a non-Archimedean absolute value, for example $K=\mathbb{C}((t))$. An Abelian variety $A$ over $K$ is called *maximally degenerate* if it admits an analytic uniformisation by a torus, i.e. if there exists an isomorphism of analytic varieties $(\mathbb{G}_m^n)^{an}/\Lambda \cong A^{an}$ where $\Lambda$ is a lattice (a free subgroup of $\mathbb{G}_m^n$ which maps injectively into its tropicalization). Being maximally degenerate can more generally be defined as having a skeleton in the sense of Berkovich of dimension coinciding with the dimension of the variety.

Elliptic curves that are maximally degenerate are characterized as those having j-invariant of absolute value $> 1$. How can one describe the set of points in the moduli space of curves of genus $g \geq 2$, $\mathcal{M}_g(K)$, which corresponds to curves which have a maximally degenerate Jacobian? Is this set semi-algebraic?