It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with weights $4$ and $6$. As far as I can tell, this is more or less directly linked to the fact that the ring of holomorphic modular forms for $SL_2(\mathbb Z)$ is a polynomial ring with generators $E_4$ and $E_6$. As a consequence, $\overline M_{1,1}$ is a smooth toric Deligne-Mumford stack.

It has been shown in Eichler-Zagier's book

http://carlossicoli.free.fr/E/Eichler_M.,_Zagier_D.-The_theory_of_Jacobi_forms.pdf

that the ring of weak Jacobi forms of even weight is a polynomial ring with a double grading by weight and index (see Theorem 9.3 on page 108). The bigradings of the variables are $(4,0),(6,0),(-2,1),(0,1)$. It is thus tempting to consider a GIT quotient of $\mathbb C^4$ with respect to the corresponding action of $\mathbb (C^*)^2$ by looking at the Gale dual of the set of weights. This would be a smooth toric DM stack of dimension two and Picard number 2. Dimension two makes a choice of triangulation automatic.

Does one get the universal elliptic curve $\overline M_{1,2}$ this way? If so, is there a reference for this? My guess is that it is a little bit off, but I am not positive. (Note: I believe the fibers of the projection map to $\overline M_{1,1}$ are elliptic curves modulo Kummer involution, which is why they are rational).