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The canonical map of a foliated surface of general type

Xin Lü

Abstract

Let $(S,\mathcal{F})$ be a foliated surface over the complex number of general type, i.e., the Kodaira dimension $\mathrm{Kod}(\mathcal{F})=2$. We study the geometry of the canonical map $\varphi$ of the foliated surface $(S,\mathcal{F})$, and prove several boundedness results on the canonical map $\varphi$, generalizing Beauville's beautiful work on the canonical maps of algebraic surfaces to foliated surfaces. As an application, we prove three Noether type inequalities for $(S,\mathcal{F})$ depending on the Kodaira dimension of the surface $S$.

The canonical map of a foliated surface of general type

Abstract

Let be a foliated surface over the complex number of general type, i.e., the Kodaira dimension . We study the geometry of the canonical map of the foliated surface , and prove several boundedness results on the canonical map , generalizing Beauville's beautiful work on the canonical maps of algebraic surfaces to foliated surfaces. As an application, we prove three Noether type inequalities for depending on the Kodaira dimension of the surface .

Paper Structure

This paper contains 14 sections, 32 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Basic definitions
  4. The birational invariants and the canonical map of a foliated surface
  5. The canonical map is generically finite
  6. The degree of surfaces in projective spaces
  7. The degree of the canonical maps and canonical foliated surfaces
  8. Examples with maximal canonical degree
  9. The canonical map induces a fibration
  10. Foliation defined by the canonical fibration
  11. Foliation different from the canonical fibration
  12. Examples of foliations with canonical fibrations
  13. The Noether type inequalities
  14. The Riemann-Hurwitz formula for foliated surfaces

Key Result

Theorem 1.1

Let $\varphi_1:\,S \dashrightarrow \Sigma \hookrightarrow \mathbb P^{p_g(S)-1}$ be canonical map of $S$.

Theorems & Definitions (85)

  • Theorem 1.1: Beauville
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1
  • Theorem 1.6
  • Remark 2
  • Definition 1
  • Definition 2
  • ...and 75 more