Birational morphisms in quantum toric geometry

TL;DR

The paper extends classical toric birational geometry to quantum toric stacks, allowing irrational cone data and continuous families of stacks. It develops a robust combinatorial framework of birational morphisms via birational fan morphisms and weighted blow-ups, providing explicit fiber and exceptional divisor descriptions in both rational and irrational settings. It then connects these birational operations to polytope and fan cobordisms, yielding a decomposition theory for birational maps and a deformation-stable perspective through moduli of quantum fans. In the rational regime, it ties these birational moves to LVM surgery and projective toric orbifolds, while the irrational framework generalizes to fan cobordisms, offering a broad, stack-theoretic account of birational transformations in quantum toric geometry.

Abstract

In this paper, we investigate birational toric morphisms between quantum toric stacks -- namely, toric (analytic) stacks associated with fans whose cones may be irrational -- focusing on two primary classes of examples: weighted blow-ups with arbitrary weights, and morphisms induced by cobordisms.

Figures (13)

• Figure 1: blow up of a plane at the origin
• Figure 2: Birational morphism from a quantum projective space
• Figure 3: Cut full triangle
• Figure 4: Cobordism between the triangle and the cut triangle
• Figure 5: Initial and final polytope

Theorems & Definitions (86)

• Definition 1.1.1
• Definition 1.1.2
• Remark 1.1.3
• Definition 1.1.4
• Example 1.1.5
• Definition 1.1.6
• Theorem 1.1.7: katzarkov2020quantum Theorem 6.24, boivin2020nonsimplicial Theorem 4.2.2.2
• Remark 1.1.8
• Remark 1.1.9
• Theorem 1.1.10: katzarkov2020quantum Theorem 7.6, Quantum GIT