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Equivalent genus-2 factorizations of type (4, 3)

Evan Huang

Abstract

The genus-2 fibrations of type (4, 3) found by Baykur-Korkmaz, Hamada, and Xiao are supported on the same total space. In this short note, we show that the Lefschetz fibration structures are the same.

Equivalent genus-2 factorizations of type (4, 3)

Abstract

The genus-2 fibrations of type (4, 3) found by Baykur-Korkmaz, Hamada, and Xiao are supported on the same total space. In this short note, we show that the Lefschetz fibration structures are the same.
Paper Structure (6 sections, 3 theorems, 12 equations, 9 figures)

This paper contains 6 sections, 3 theorems, 12 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Lefschetz fibrations
  3. Computations in $S_{0,6}$
  4. Equivalences
  5. Reverse engineering Hamada's $(4,3)$ factorization
  6. Vanishing cycles and Hurwitz moves

Key Result

Theorem 1.1

The Lefschetz fibrations corresponding to the $(4,3)$ factorizations of Baykur-Korkmaz, Hamada, and Xiao are all isomorphic.

Figures (9)

  • Figure 1: The standard chain of curves on $\Sigma_2$.
  • Figure 2: Sections of $(4,3)$ factorizations.
  • Figure 3: Vanishing cycles in Baykur-Korkmaz's factorization.
  • Figure 4: Vanishing cycles in Hamada's factorization.
  • Figure 5: Vanishing cycles of Xiao's $(4,3)$ fibration. Sections are in red, see Remark \ref{['rmk:red_sections']}.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 3.1
  • Proposition 3.1
  • Remark 4.1