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

TL;DR

This paper constructs a mirror symmetry-inspired compactification of the moduli space of toric pairs, linking the secondary fan to the Mori fan of the mirror family, providing an explicit normalized compactification.

Contribution

It introduces a new compactification of the moduli of toric pairs using mirror symmetry, connecting the secondary fan to the Mori fan of the mirror family.

Findings

Verified the secondary fan equals the Mori fan of the mirror family.

Constructed an explicit normalized compactification of the moduli space.

Connected mirror symmetry predictions with moduli space geometry.

Abstract

We construct a compactification of the moduli of toric pairs by using ideas from mirror symmetry. The secondary fan $\Sigma(Q)$ 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 $\Sigma(Q)$ 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 $\mathscr{T}_Q$ of the moduli of toric pairs which is the normalization of the compactification in [Ale02] and [Ols08].