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Smoothings from zero mutable Laurent polynomials via log resolutions and divisorial extractions

Tim Gräfnitz

TL;DR

The paper develops a program to realize smoothings of affine Gorenstein toric 3-folds by constructing log crepant resolutions from compatible divisorial extractions dictated by zero mutable Laurent polynomials (ZMLPs). It establishes a mutation-theoretic framework for ZMLPs on polygons, especially rectangular triangles, and provides explicit classifications and mutation chains (Tom, Jerry, Spike, Tyke) that guide the construction of log structures from toric degenerations. By pairing ZMLP divisibility data with central subdivisions of dual cones and joint log-compatibility conditions, the authors outline a path from combinatorial data to toroidal log smoothings and eventual smoothings via generalized FFR techniques. The results yield concrete instances where the CFP conjecture's smoothing components correspond to specific ZMLPs, and deliver explicit, compatible divisorial extractions for several infinite families, advancing the program to prove the conjecture in the affine cone setting and enriching the link between mirror-symmetric mutations and toric degenerations.

Abstract

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log crepant log resolutions constructed from compatible collections of divisorial extractions. For affine cones over weighted projective planes we prove for several infinite families of zero mutable Laurent polynomials that they indeed describe curves that admit a compatible collection of divisorial extractions. The construction of log crepant log resolutions and smoothings will be worked out in joint work with Alessio Corti and Helge Ruddat.

Smoothings from zero mutable Laurent polynomials via log resolutions and divisorial extractions

TL;DR

The paper develops a program to realize smoothings of affine Gorenstein toric 3-folds by constructing log crepant resolutions from compatible divisorial extractions dictated by zero mutable Laurent polynomials (ZMLPs). It establishes a mutation-theoretic framework for ZMLPs on polygons, especially rectangular triangles, and provides explicit classifications and mutation chains (Tom, Jerry, Spike, Tyke) that guide the construction of log structures from toric degenerations. By pairing ZMLP divisibility data with central subdivisions of dual cones and joint log-compatibility conditions, the authors outline a path from combinatorial data to toroidal log smoothings and eventual smoothings via generalized FFR techniques. The results yield concrete instances where the CFP conjecture's smoothing components correspond to specific ZMLPs, and deliver explicit, compatible divisorial extractions for several infinite families, advancing the program to prove the conjecture in the affine cone setting and enriching the link between mirror-symmetric mutations and toric degenerations.

Abstract

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log crepant log resolutions constructed from compatible collections of divisorial extractions. For affine cones over weighted projective planes we prove for several infinite families of zero mutable Laurent polynomials that they indeed describe curves that admit a compatible collection of divisorial extractions. The construction of log crepant log resolutions and smoothings will be worked out in joint work with Alessio Corti and Helge Ruddat.

Paper Structure

This paper contains 19 sections, 33 theorems, 59 equations, 16 figures, 2 tables.

Table of Contents

  1. Zero mutable Laurent polynomials
  2. ZMLPs on polygons: divisibility steps
  3. ZMLPs on rectangular triangles
  4. Triangular mutations
  5. Classification results: Tom, Jerry, Spike, Tyke and the like
  6. Large triangles
  7. Toric degenerations and log smoothings
  8. Toric degenerations
  9. Log structures
  10. Log structures with prescribed singular locus
  11. Log structures from zero mutable Laurent polynomials
  12. Divisorial extractions and log resolutions
  13. Constructing compatible divisorial extractions
  14. Rectangular triangles: reduction to $\mathbb{A}^3$
  15. $A_n$-triangles
  16. ...and 4 more sections

Key Result

Theorem 1

In all cases above, there is a chain of mutations $f \mapsto f_r \mapsto \ldots \mapsto f_1 \mapsto f_0=1$ such that all steps $f_i$ are zero mutable Laurent polynomials on primitive rectangular triangles.

Figures (16)

  • Figure 1: Zero-mutable Laurent polynomials for Tom and Jerry.
  • Figure 2: Log singular loci for Tom and Jerry.
  • Figure 3: A mutation graph showing mutation relationships between zero mutable Laurent polynomials (ZMLPs) with small Netwon polytopes.
  • Figure 4: Sequences of mutations showing that Tom and Jerry is zero mutable.
  • Figure 5: Zero mutable Laurent polynomials for Tom (left) and Jerry (right) with (required) divisibility tuples (green) and divisibility steps (red).
  • ...and 11 more figures

Theorems & Definitions (119)

  • Definition
  • Conjecture 1: CFP, Conjecture A
  • Conjecture 2
  • Conjecture 3: c.f. CGR, Conjecture 18
  • Definition
  • Example : Tom and Jerry
  • Theorem 1: Propositions \ref{['prop:a']}, \ref{['prop:b']}, \ref{['prop:d']}, \ref{['prop:c']}
  • Theorem 2: Propositions \ref{['prop:maindivex']}, \ref{['prop:An']}, \ref{['prop:tom1']}, \ref{['prop:tom2']}, \ref{['prop:jerrytyke']}
  • Conjecture 4
  • Definition 1.1
  • ...and 109 more