Abstract:

Mori invented a technique called as Bend and Break lemma which claims that if we deform a curve with fixed points, then it breaks into the union of several curves such that some of them are rational. This technique has wide applications ranging from rationally connectedness of smooth Fano varieties, Cone theorem for smooth projective varieties, to boundedness of smooth Fano varieties. However, a priori there is no control on breaking curves so in particular, an outcome of Bend and Break could be a singular point of the moduli space of rational curves. With Brian Lehmann, we propose Movable Bend and Break conjecture which claims that a free rational curve of enough high degree can degenerate to the union of two free rational curves in the moduli space of stable maps, and we confirm this conjecture for sections of del Pezzo fibrations over an arbitrary smooth projective curve. In this talk I will explain some of ideas of the proof of MBB for del Pezzo fibrations as well as its applications to Batyrev’s conjecture and Geometric Mann’s conjecture. This is joint work with Brian Lehmann.