Abstract: The talk will introduce the F-conjecture, which states that the Mori cone of the moduli space of genus g stable curves ‾Mg,n is generated by 1-dimensional strata. We will see that the question can be formulated on any Hassett space and has a positive answer when the universal family is a P1-bundle. In particular, we review how these particular Hassett spaces are naturally isomorphic to certain GIT quotients (P1)n // PGL2, and we characterize these spaces as the targets of the birational contractions of Bln-1Pn-3 with Q-factorial image.