Normal hyperplane sections of normal schemes in mixed characteristic

Abstract

The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck-Lefschetz type in mixed characteristic. Dedicated to Prof. Gennady Lyubeznik on the occasion of his 60th birthday.The proof of the theorem is obtained as an immediate corollary of the Bertini type theorem for normal schemes in mixed characteristic. The hypothesis that A is unramified is forced in the proof of the local Bertini theorem in [15], where we needed to estimate

Bertini theorems revisited

“…The normal crossing special fiber condition was recently removed by Binda–Krishna [2] under the condition that the residue field is infinite and perfect. A form of Bertini‐normality theorem for affine and flat normal schemes over a discrete valuation ring was obtained by Horiuchi–Shimomoto [24] under some conditions on the ring. To our knowledge, apart from the above results, no other Bertini type result seems to be known for schemes over a discrete valuation ring.…”