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Gröbner Fans and Minimal Embedded Toric Resolutions of Rational Double Point Singularities

Büşra Karadeniz Şen

TL;DR

The paper develops a framework to obtain minimal embedded toric resolutions of $ADE$-singularities by explicitly constructing and refining Gröbner fans via the profile of cones. It shows that for Newton non-degenerate ADE hypersurfaces, regular refinements of the Gröbner fan compatible with the fan yield proper birational morphisms with smooth, transversal exceptional divisors, and that the skeleton of the Gröbner fan recovers the minimal resolution graph. For each type $A_n$, $D_n$, $E_6$, $E_7$, $E_8$, the authors compute the Gröbner fans, determine profiles of maximal cones, and extract lattice points inside the profiles to generate irreducible refinement vectors, producing minimal embedded toric resolutions. They also connect these refinements to jet-scheme components, showing that the number of irreducible jet components matches the vertices of the minimal resolution graph in large jet levels, thereby linking combinatorial, geometric, and jet-theoretic data.

Abstract

In [19], the authors give minimal embedded toric resolutions of ADE-singularities in C^3 by constructing regular refinements of their dual Newton polyhedrons with the elements of their embedded valuation sets derived from the jet schemes constructed in [18]. On the other hand in [1] and [2], the authors represent the Gröbner fan of a Newton non-degenerate variety and prove that a regular refinement of the Gröbner fan of such a singularity yields an embedded toric resolution. In this paper, we reconstruct embedded toric resolutions of ADE-singularities: We give the explicit constructions of their Gröbner fans and refine them using the concept of profile.

Gröbner Fans and Minimal Embedded Toric Resolutions of Rational Double Point Singularities

TL;DR

The paper develops a framework to obtain minimal embedded toric resolutions of -singularities by explicitly constructing and refining Gröbner fans via the profile of cones. It shows that for Newton non-degenerate ADE hypersurfaces, regular refinements of the Gröbner fan compatible with the fan yield proper birational morphisms with smooth, transversal exceptional divisors, and that the skeleton of the Gröbner fan recovers the minimal resolution graph. For each type , , , , , the authors compute the Gröbner fans, determine profiles of maximal cones, and extract lattice points inside the profiles to generate irreducible refinement vectors, producing minimal embedded toric resolutions. They also connect these refinements to jet-scheme components, showing that the number of irreducible jet components matches the vertices of the minimal resolution graph in large jet levels, thereby linking combinatorial, geometric, and jet-theoretic data.

Abstract

In [19], the authors give minimal embedded toric resolutions of ADE-singularities in C^3 by constructing regular refinements of their dual Newton polyhedrons with the elements of their embedded valuation sets derived from the jet schemes constructed in [18]. On the other hand in [1] and [2], the authors represent the Gröbner fan of a Newton non-degenerate variety and prove that a regular refinement of the Gröbner fan of such a singularity yields an embedded toric resolution. In this paper, we reconstruct embedded toric resolutions of ADE-singularities: We give the explicit constructions of their Gröbner fans and refine them using the concept of profile.

Paper Structure

This paper contains 9 sections, 26 theorems, 61 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Profile of a Cone
  3. Newton Non-Degeneracy and Gröbner Fan
  4. Embedded Toric Resolution
  5. $A_n$-singularities
  6. $D_n$-singularities
  7. $E_6$-singularity
  8. $E_7$-singularity
  9. $E_8$-singularity

Key Result

Theorem 1.1

AGSar-hu Let $I\subset \mathbb{C}[x_1,x_2,\ldots,x_n]$ be a Newton non-degenerate ideal and let $X=V(I)\subset X_{C}$ where $X_{C}$ is the toric variety associated to the cone $C$. Let $\Sigma$ be a regular refinement of $C$ which is compatible with Gröbner fan of $X$. Then the associated toric morp

Figures (22)

  • Figure 1: The Gröbner fan and its Gröbner cones for $A_n$-singularities
  • Figure 2: The profiles of maximal dimensional Gröbner cones for $A_n$-singularities
  • Figure 3: A regular refinement for $A_n$-singularities
  • Figure 4: The resolution graph of $A_n$-singularities
  • Figure 5: The graph of $A_n$-singularities obtaining from the construction
  • ...and 17 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Remark 4.1
  • Definition 4.2
  • Remark 4.4
  • Theorem 4.5
  • proof
  • ...and 39 more