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Generically log smooth families via generators and relations

Simon Felten

TL;DR

This work develops an algorithmic toolkit for studying generically log smooth degenerations of affine and projective families by encoding the total space with a generators-and-relations presentation and leveraging log geometry. It introduces an explicit, computable GLS criterion (including dkink and Ext^1-based obstructions) that certifies when a flat family is generically log smooth, and it generalizes the framework to toroidal crossing schemes with a refined classifying map via $\eta_V^{(\ell)}$. Central results include a differential-log-smoothness perspective, finite determinacy of log structures on the central fiber, and a tight correspondence between GLS data and toroidal crossing structures, enabling practical computations (and a Macaulay2 implementation). The theory yields both affine and projective variants, provides practical checks using infinitesimal thickenings, and establishes how log structures on the central fiber can be compared or identified across families through kink data. Collectively, these contributions offer a robust, implementable approach for analyzing log singularities and smoothing degenerations in explicit examples, with potential impact on log Gromov–Witten theory and mirror symmetry constructions.

Abstract

Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\barρ)$.

Generically log smooth families via generators and relations

TL;DR

This work develops an algorithmic toolkit for studying generically log smooth degenerations of affine and projective families by encoding the total space with a generators-and-relations presentation and leveraging log geometry. It introduces an explicit, computable GLS criterion (including dkink and Ext^1-based obstructions) that certifies when a flat family is generically log smooth, and it generalizes the framework to toroidal crossing schemes with a refined classifying map via . Central results include a differential-log-smoothness perspective, finite determinacy of log structures on the central fiber, and a tight correspondence between GLS data and toroidal crossing structures, enabling practical computations (and a Macaulay2 implementation). The theory yields both affine and projective variants, provides practical checks using infinitesimal thickenings, and establishes how log structures on the central fiber can be compared or identified across families through kink data. Collectively, these contributions offer a robust, implementable approach for analyzing log singularities and smoothing degenerations in explicit examples, with potential impact on log Gromov–Witten theory and mirror symmetry constructions.

Abstract

Let be an affine flat morphism of finite type, and let . Then, we obtain a morphism of log schemes . In this article, we develop algorithmic tools to study the log-geometric properties of by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf of log smooth structures on a toroidal crossing scheme .
Paper Structure (33 sections, 38 theorems, 103 equations, 2 figures)

This paper contains 33 sections, 38 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An algorithmic criterion for generic log smoothness
  3. Toroidal crossing schemes
  4. Differential log smoothness
  5. Applications
  6. Acknowledgments
  7. Results and examples
  8. Generic log smoothness: main results
  9. Generically log smooth families
  10. The criterion for generic log smoothness
  11. The criterion in explicit affine families
  12. The criterion in explicit projective families
  13. The infinitesimal criterion for generic log smoothness
  14. Toroidal crossing schemes: main results
  15. Toroidal crossing schemes
  16. ...and 18 more sections

Key Result

Proposition 1.3

Let $\Bbbk$ be an algebraically closed field, and let $V/\Bbbk$ be reduced, separated, of finite type, and assume that every irreducible component has the same dimension $d \geq 1$. Let $\nu\colon \widetilde{V} \to V$ be the normalization map, and let $C \subset V$ be the conductor locus. Let $\mu\c

Figures (2)

  • Figure 1: The central fiber in Example \ref{['ex:112']}
  • Figure 2: The central fiber in Example \ref{['ex:projective-112']}

Theorems & Definitions (126)

  • Example 1.1
  • Example 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Example 1.5
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.7
  • Remark 2.8
  • Lemma 2.10
  • ...and 116 more