Natural compactification of the moduli of toric pairs from the perspective of mirror symmetry

We construct a compactification of the moduli of toric pairs by using ideas from mirror symmetry. The secondary fan $\mathrm{\Sigma }(Q)$ is used in [Ale02] to parametrize degenerations of toric pairs. It is also used in [CLS11] to control the variation of GIT. We verify the prediction of mirror symmetry that $\mathrm{\Sigma }(Q)$ for the moduli of toric pairs is equal to the Mori fan of the relative minimal models of the mirror family. As a result, we give an explicit construction of the compactification ${\mathcal{T}}_Q$ of the moduli of toric pairs which is the normalization of the compactification in [Ale02] and [Ols08].