  We show that the ring of Siegel-Jacobi forms of fixed degree and of fixed or
bounded ratio between weight and index is not finitely generated. Our main tool
is the theory of toroidal b-divisors and their relation to convex geometry. As
a byproduct of our methods, we prove a conjecture of Kramer about the
representation of all Siegel-Jacobi forms as sections of certain line bundles
and we recover a formula due to Tai for the asymptotic dimension of the space
of Siegel-Jacobi forms of given ratio between weight and index.
