June 06, 2017
2:00 PM - 3:30 PM
Math Tower 5-127
|Dingxin Zhang, Stony Brook University|
Degeneration of slopes
A. Grothendieck proves that the Newton polygons of a family of smooth projective algebraic varieties defined on a field of characteristic p > 0 go up under a smooth specialization. When the family acquires semistable singular members, we prove the smallest slope of the Newton polygons attached to the rigid cohomology groups cannot become smaller upon degeneration. This is achieved by constructing the “generic higher direct images” for a singular morphism, using the convergent topoi. Our result generalizes the theorem of Grothendieck, and partially answers a question of H. Esanult. As geometric applications, we prove that several families of algebraic varieties must have “ordinary” members. Special cases of these applications recover older theorems of L. Illusie and Wolf-Riedl.