The canonical map of a foliated surface of general type

TL;DR

This paper investigates the geometry of the canonical map of complex foliated surfaces of general type, extending classical results to the foliated setting and establishing boundedness and inequality results.

Contribution

It generalizes Beauville's work on algebraic surfaces to foliated surfaces, providing boundedness results and Noether type inequalities based on Kodaira dimension.

Findings

Boundedness results for the canonical map of foliated surfaces.

Generalization of Beauville's work to foliated surfaces.

Three Noether type inequalities depending on Kodaira dimension.

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 $S$.