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Classification of normal toric surfaces resolved by a single Nash blowup

Amador Cruz-Fuentes

TL;DR

The paper provides a complete CF-based classification of normal toric surfaces resolvable by a single Nash blowup (normalized and non-normalized). It develops a combinatorial framework linking continued fractions, Newton polyhedra, and affine charts of Nash blowups, and derives explicit necessary-and-sufficient CF forms for smoothness in both the normalized and non-normalized cases (the latter over char 0). The results show that resolution in one step is tightly governed by the pattern of consecutive CF coefficients, with concrete lists of allowed forms up to long continued fractions. This advances understanding of toric singularities and clarifies where single-step Nash approaches succeed or fail in low dimensions.

Abstract

We present a complete classification of normal toric surfaces that are resolved by a single normalized Nash blowup. Likewise, we obtain a complete classification of those resolved by a single Nash blowup. In both cases, the classification is expressed in terms of the continued fraction associated with the normal toric surface.

Classification of normal toric surfaces resolved by a single Nash blowup

TL;DR

The paper provides a complete CF-based classification of normal toric surfaces resolvable by a single Nash blowup (normalized and non-normalized). It develops a combinatorial framework linking continued fractions, Newton polyhedra, and affine charts of Nash blowups, and derives explicit necessary-and-sufficient CF forms for smoothness in both the normalized and non-normalized cases (the latter over char 0). The results show that resolution in one step is tightly governed by the pattern of consecutive CF coefficients, with concrete lists of allowed forms up to long continued fractions. This advances understanding of toric singularities and clarifies where single-step Nash approaches succeed or fail in low dimensions.

Abstract

We present a complete classification of normal toric surfaces that are resolved by a single normalized Nash blowup. Likewise, we obtain a complete classification of those resolved by a single Nash blowup. In both cases, the classification is expressed in terms of the continued fraction associated with the normal toric surface.

Paper Structure

This paper contains 7 sections, 14 theorems, 64 equations.

Table of Contents

  1. Nash blowups and toric varieties
  2. Continued fractions in the study of toric surfaces
  3. Combinatorial description of the Nash blowup of a toric surface
  4. Vertices of the Newton polyhedron
  5. One-step resolution
  6. One-step resolution via normalized Nash blowup
  7. One-step resolution via Nash blowup

Key Result

Theorem A

Let $\operatorname{char}(\mathbb{K}) = p \ge 0$. The normalized Nash blowup of a normal toric surface is smooth if and only if its associated continued fraction $[1, a_2, \dots, a_r]$ is one of the following:

Theorems & Definitions (31)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: GonzalezSprinberg1977ventailsEDGTDJN
  • Theorem 1.4: GonzalezSprinberg1977ventailsEDGTDJN
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 21 more