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Log Canonical Thresholds for Plane Curves in Arbitrary Characteristic

Chih-Kuang Lee

TL;DR

This paper extends the computation of the log canonical threshold for plane curves from characteristic zero to arbitrary characteristic by a valuation-theoretic approach. It introduces and leverages the valuative tree $\mathcal{V}$, sequences of key polynomials (SKPs), and the universal dual graph $\Gamma$ to connect birational data with semivaluations, enabling a direct calculation of $lct$ via $lct_o(f)=\frac{1}{v_f(x)}+\frac{1}{v_f(y)}$ for irreducible $f$ with a tangent cone of the prescribed form. A central contribution is establishing an isomorphism between $\Gamma$ and $\mathcal{V}$ that preserves multiplicities in arbitrary characteristic, together with a Newton-polyhedron-based analysis that yields the needed inequality in the main theorem. The results provide a robust, valuation-theoretic framework for computing LCTs of plane curves over fields of any characteristic, with potential broader impact on singularity theory and birational geometry.

Abstract

We generalize the formula for the log canonical threshold(LCT) of plane curves over the complex numbers to arbitrary characteristics. Our proof relies purely on valuation theory, instead of on the theory of $D$-modules.

Log Canonical Thresholds for Plane Curves in Arbitrary Characteristic

TL;DR

This paper extends the computation of the log canonical threshold for plane curves from characteristic zero to arbitrary characteristic by a valuation-theoretic approach. It introduces and leverages the valuative tree , sequences of key polynomials (SKPs), and the universal dual graph to connect birational data with semivaluations, enabling a direct calculation of via for irreducible with a tangent cone of the prescribed form. A central contribution is establishing an isomorphism between and that preserves multiplicities in arbitrary characteristic, together with a Newton-polyhedron-based analysis that yields the needed inequality in the main theorem. The results provide a robust, valuation-theoretic framework for computing LCTs of plane curves over fields of any characteristic, with potential broader impact on singularity theory and birational geometry.

Abstract

We generalize the formula for the log canonical threshold(LCT) of plane curves over the complex numbers to arbitrary characteristics. Our proof relies purely on valuation theory, instead of on the theory of -modules.
Paper Structure (11 sections, 16 theorems, 54 equations)

This paper contains 11 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Valuation Theory
  3. Semivaluations
  4. Log Canonical Thresholds (LCTs)
  5. Non-metric Trees
  6. The Valuative Tree V
  7. Sequences of Key Polynomials (SKPs)
  8. Parametrizations on V
  9. The Universal Dual Graph
  10. Newton Polyhedra of Formal Power Series
  11. The Main Theorem

Key Result

Theorem 1.1

Let ${f}\in\Bbbk[[x,y]]$ be irreducible, and assume that the tangent cone of $V({f})$ is defined by the equation $y^{\operatorname{ord}_\mathfrak{m}({f})} = 0$. Write $v_{{f}}$ for the corresponding non-normalized curve semivaluation. Then, we have

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 2.1: FavreJonsson2004
  • Lemma 2.3: First Properties of SKPs
  • Proposition 2.4: cf. FavreJonsson2004
  • Proposition 2.5
  • Theorem 2.6: The Isomorphism between $\Gamma$ and $\mathcal{V}$
  • Definition 2.7: $\mathcal{V}_{\textup{div}}$
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 21 more