The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan. The next obvious step in this endeavour, in analogy with the classical case of projective varieties and log pairs, is to construct moduli spaces for foliated varieties (starting from the general type case). The first question to ask, on the road towards constructing such a moduli space, is how to show that foliated varieties of fixed Kodaira dimension are bounded, that is, they come in finitely many algebraic families ― provided, of course, that we fix certain appropriate numerical invariants. It turns out that, to best answer this question, rather than working with the canonical divisor of a foliation it is better to consider linear systems of the form $|nK_X + mK_F|, n,m >0, as those encode a lot of the positivity features that classically the canonical divisor of a projective variety displays.
In this talk, I will introduce this framework and explain how this approach leads to answering the question about boundedness for foliated surfaces. Time permitting, I will address also what happens when we try to construct a moduli functor, or rather, what we have been finding out, so far. This talk features joint work with C. Spicer, work with J. Pereira, and work in progress with M. McQuillan and C. Spicer.