2018/10/15 by Christensen, Atticus
#14C22 (Primary) 14F30 #14G17 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1810.06550
André and Maulik--Poonen proved that for any smooth proper family X→ B of varieties over an algebraically closed field of characteristic 0, there is a closed fiber whose Néron-Severi group has the same rank as that of the Néron-Severi group of the geometric generic fiber. We prove the analogous statement over algebraically closed fields of characteristic p>0 which are not isomorphic to \mathbbFp. Furthermore, we prove that for any algebraically closed field k of characteristic p>0 and smooth proper family X→ B of k-varieties, there exists a dense open subvariety U⊆ B and integer N such that for each map x:Spec k[[t]]→ U, the p-torsion in the cokernel of the specialization map from the Néron-Severi group of the pullback of X to the geometric generic fiber of x to the Néron-Severi group of the pullback of X to the special fiber of x is killed by pN. Finally, we prove that for a curve C over k and family \mathscrX→\mathscrB of smooth C-schemes, there exists a dense Zariski open U⊆ \mathscrB such that for a local uniformizer t at any closed point of C, the rank of the Néron-Severi group jumps only on a t-adic nowhere dense set t. The crystalline Lefschetz (1,1) theorem of Morrow is a key ingredient in the proofs.