Gluing moduli spaces of quantum toric stacks via secondary fan
Antoine Boivin
TL;DR
The paper extends the moduli theory of toric objects to quantum toric stacks by developing a quantum secondary-fan framework that accommodates non-reductive quotients and irrational combinatorics. It adapts GKZ-style secondary fans to parameterize wall-crossings between moduli spaces of quantum toric stacks, yielding a unified augmented moduli space whose connectedness links quantum fans to quantum projective spaces. A universal family over the secondary fan is constructed, and a gluing procedure across combinatorial types leads to a compactified, thickened big moduli stack $\mathscr{K}(d,n)$ that encodes all deformations and degenerations. The results describe wall-crossings, flips, and cobordisms in the quantum setting, and establish a canonical framework for augmented moduli spaces and their compactifications with potential analogues of LVMB cobordisms.
Abstract
The extension from toric varieties to quantum toric stacks allows for the study of moduli spaces of toric objects with fixed combinatorial structures, as we now consider general finitely generated subgroups of $\mathbb{R}^n$ as "lattices." This paper aims to construct a moduli space that encompasses all such moduli spaces for a given dimension of the ambient space. To achieve this, we adapt the construction of the secondary fan within the quantum framework. This approach provides a description of wall-crossings between different moduli spaces, analogous to those observed in LVMB manifolds.