Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singular Log Structures and Log Crepant Log Resolutions I

Alessio Corti, Tim Graefnitz, Helge Ruddat

TL;DR

The paper proposes a new framework of zero-mutable log structures in dimension $3$ and conjectures that their log crepant resolutions exist in a projective setting, enabling a log-birational approach to smoothing singularities arising from toric Fano deformations. It develops a mutation-based description of log data, provides explicit A$_n$, Tom, and Jerry examples, and demonstrates how to construct log crepant resolutions within this framework, highlighting both potential and limitations. The authors connect their conjectures to the deformation theory studied in MR4381899 and to the Gross--Siebert program, outlining a strategy to obtain distinguished smoothings via scattering diagrams and KS/Gross--Siebert techniques, with a view toward log Gromov--Witten theory of the resolutions. They sketch higher-dimensional extensions via two steps (Step I and Step II), defining how the framework could generalize to larger lattices and polytopes while preserving the core ideas of wall functions, mutations, and log crepant resolutions.

Abstract

We introduce a class of singular log schemes in three dimensions and conjecture that log schemes in this class admit log crepant log resolutions. We provide examples as evidence and relate this conjecture to the conjecture made in [4] and the Gross--Siebert program.

Singular Log Structures and Log Crepant Log Resolutions I

TL;DR

The paper proposes a new framework of zero-mutable log structures in dimension and conjectures that their log crepant resolutions exist in a projective setting, enabling a log-birational approach to smoothing singularities arising from toric Fano deformations. It develops a mutation-based description of log data, provides explicit A, Tom, and Jerry examples, and demonstrates how to construct log crepant resolutions within this framework, highlighting both potential and limitations. The authors connect their conjectures to the deformation theory studied in MR4381899 and to the Gross--Siebert program, outlining a strategy to obtain distinguished smoothings via scattering diagrams and KS/Gross--Siebert techniques, with a view toward log Gromov--Witten theory of the resolutions. They sketch higher-dimensional extensions via two steps (Step I and Step II), defining how the framework could generalize to larger lattices and polytopes while preserving the core ideas of wall functions, mutations, and log crepant resolutions.

Abstract

We introduce a class of singular log schemes in three dimensions and conjecture that log schemes in this class admit log crepant log resolutions. We provide examples as evidence and relate this conjecture to the conjecture made in [4] and the Gross--Siebert program.

Paper Structure

This paper contains 13 sections, 2 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Conjecture
  3. Log data
  4. Mutations of log data
  5. Zero-mutable log structures and statement of the Conjecture
  6. Examples
  7. What happens in codimension $1$
  8. The $A_n$ singularity
  9. Tom and Jerry
  10. Relation to MR4381899 and the Gross--Siebert program
  11. Higher dimensions
  12. Step I
  13. Step II

Key Result

Lemma 8

The set $\mathop{\mathrm{LS}}\nolimits_{k^\dagger}(X)$ of log structures on $X$ over $k^\dagger$ compatible with the gtc structure is the set of data consisting of

Figures (6)

  • Figure 1: We represent a log datum by its polygon with each edge labelled by the corresponding $(e_i,\nu_i)$. The figure shows a mutation in $((-2,0),(2))$.
  • Figure 2: Sketch of the log singular locus for the $A_3$ log structure represented inside the fan $\Sigma$, labelling the lattice points that give Stanley Reisner ring generators.
  • Figure 3: Sketch of the central fiber $t=0$ inside the intermediate space $\mathfrak{Y}'$ in the IIIa resolution of the $A_3$ log structure. The inverse image of $Z_1$ and $Z_3$ gives a $\mathbb{P}^1$-bundle respectively while the strict transform of $Z_2$ is a disjoint union of four lines, so a smooth curve.
  • Figure 4: Tom log datum and mutation to the $A_1$ log datum with $\ell_{j,k}$ in bold.
  • Figure 5: Jerry log datum and mutation to the $A_2$ log datum with $\ell_{j,k}$ in bold.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Remark 7
  • Lemma 8
  • Definition 9
  • proof
  • Remark 11
  • Definition 12
  • ...and 9 more