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Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields

Charles Li

Abstract

An exposition on Spivakovsky's dual graphs of valuations on function fields of dimension two is first given, leading to a proof of minimal generating sequences for the non-divisorial valuations. It should be noted that the definition of generating sequence used in this paper is different from Spivakovsky's original usage. This change leads to an explicit formulation of generating sequence values for the non-divisorial cases in terms of data from their dual graphs. The proofs are elementary in the sense that only continued fractions and the linear Diophantine Frobenius problem from classical number theory are used.

Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields

Abstract

An exposition on Spivakovsky's dual graphs of valuations on function fields of dimension two is first given, leading to a proof of minimal generating sequences for the non-divisorial valuations. It should be noted that the definition of generating sequence used in this paper is different from Spivakovsky's original usage. This change leads to an explicit formulation of generating sequence values for the non-divisorial cases in terms of data from their dual graphs. The proofs are elementary in the sense that only continued fractions and the linear Diophantine Frobenius problem from classical number theory are used.

Paper Structure

This paper contains 5 sections, 19 theorems, 127 equations, 17 figures.

Table of Contents

  1. Introduction
  2. dual graphs
  3. generating sequences
  4. proofs
  5. concluding remarks

Key Result

Proposition 2.3

Let $\lambda_{-1}=0$, $\lambda_0=1$, $\mu_{-1}=1$ and $\mu_0=0$ by convention. For $i\geq 1$, we have the basic recursive formulas:

Figures (17)

  • Figure 1: The first four stages in building the dual graph
  • Figure 2: After the fifth blowup
  • Figure 3: After the sixth blowup
  • Figure 4: After the sixth blowup, again
  • Figure 5: After the seventh blowup
  • ...and 12 more figures

Theorems & Definitions (64)

  • Example 2.1
  • Definition
  • Definition
  • Remark
  • Definition
  • Remark
  • Definition
  • Remark
  • Example 2.2
  • Remark
  • ...and 54 more