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We show that the ring of Siegel-Jacobi forms of fixed degree and of fixed orbounded ratio between weight and index is not finitely generated. Our main toolis the theory of toroidal b-divisors and their relation to convex geometry. Asa byproduct of our methods we recover a formula due to Tai for the asymptoticdimension of the space of Siegel-Jacobi forms of given ratio between weight andindex.

Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated