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Birational description of moduli spaces of rank 2 logarithmic connections

Takafumi Matsumoto

Abstract

In this paper, we provide an explicit description of the Zariski-open subset of the moduli space of rank 2 parabolic logarithmic connections in the case $g\geq 2$. Our approach is to analyze the underlying parabolic bundles and the apparent singularities of the parabolic connections. We prove that a Zariski-open subset of the product of a projective space and the moduli space of parabolic bundles gives a Darboux coordinate for the moduli space of parabolic connections.

Birational description of moduli spaces of rank 2 logarithmic connections

Abstract

In this paper, we provide an explicit description of the Zariski-open subset of the moduli space of rank 2 parabolic logarithmic connections in the case . Our approach is to analyze the underlying parabolic bundles and the apparent singularities of the parabolic connections. We prove that a Zariski-open subset of the product of a projective space and the moduli space of parabolic bundles gives a Darboux coordinate for the moduli space of parabolic connections.

Paper Structure

This paper contains 18 sections, 27 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Parabolic bundles
  3. Parabolic bundles and $\boldsymbol{\alpha}$-stability
  4. The moduli space of rank 2 quasi-parabolic bundles
  5. Parabolic connections
  6. Parabolic connections and $\boldsymbol{\alpha}$-stability
  7. The moduli space of rank 2 parabolic connections
  8. Elementary transformations
  9. The relation between parabolic weights and the birational types of moduli spaces
  10. Parabolic homomorphisms
  11. Extensions of parabolic connections
  12. Parabolic connections becoming unstable when a wall is crossed
  13. The birational structure of moduli spaces of parabolic connections
  14. The distinguished open subset of the moduli space of parabolic bundles
  15. Apparent map
  16. ...and 3 more sections

Key Result

Theorem 1.1

(Theorem 5.6 and Proposition 5.11) Under the condition (1), assume that $d=2g-1$, $\sum_{i=1}^{n}\nu^-_i \neq 0$ and $\sum_{i=1}^n(\alpha^{(i)}_2-\alpha^{(i)}_1)<1$. Then the map is an isomorphism. Hence, the rational map is birational. Moreover, $\textrm{App}$ and $\textrm{Bun}$ are Lagrangian fibrations.

Theorems & Definitions (60)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.1
  • Remark 3.2
  • ...and 50 more