Effective generation for foliated surfaces: results and applications

Abstract

We explore the birational structure and invariants of a foliated surface $(X,\mathcal{F})$ in terms of the adjoint divisor $K_{\mathcal{F}}+\epsilon K_X$, $0<\epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint general type foliated surface $(X,\mathcal{F})$, provide a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface and prove that the set of $\epsilon$-adjoint canonical models of foliations of general type and with fixed volume form a bounded family.

v2: 43 pages. Expanded and improved the exposition. Main results are unchanged; v1: 32 pages. Comments are welcome!

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