The canonical map of a foliated surface of general type

Let $(S,\mathcal{F})$ be a foliated surface over the complex number of general type, i.e., the Kodaira dimension $\mathrm{K}\mathrm{o}\mathrm{d}(\mathcal{F})=2$. We study the geometry of the canonical map $\phi$ of the foliated surface $(S,\mathcal{F})$, and prove several boundedness results on the canonical map $\phi$, 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 $S$.