  We show that a certain moduli space of minimal $A_\infty$-structures
coincides with the modular compactification $\bar{\mathcal{M}}_{1,n}(n-1)$ of
$\mathcal{M}_{1,n}$ constructed by Smyth. In addition, we describe these moduli
spaces and the universal curves over them by explicit equations, prove that
they are normal and Gorenstein, show that their Picard groups have no torsion
and that they have rational singularities if and only if $n\le 11$.
