In positive characteristic, there exist fibrations between smooth varieties where every fiber is singular or even non-reduced. In the latter case, the generic fiber of the fibration is geometrically non-reduced. We study the failure of generic smoothness by showing a generalization of Tate's genus change formula, and obtain a structural result about geometrically non-reduced varieties. Our result has applications to Fano varieties. This is joint work with Joe Waldron.

Ian Agol showed that hyperbolic groups acting geometrically on CAT(0) cube complexes are virtually special in the sense of Haglund-Wise, the last step in the proof of the virtual Haken and virtual fibering conjectures. I will talk about a generalization of this result (also obtained independently by Groves and Manning), which states that cubulated relatively hyperbolic groups are virtually special provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. In particular, we deduce virtual specialness for cubulated groups that are hyperbolic relative to virtually abelian groups, extending Wise's results for limit groups and fundamental groups of cusped hyperbolic 3-manifolds. The main ingredient of the proof is a relative version of Wise's quasi-convex hierarchy theorem, obtained using recent results by Einstein, Groves and Manning.