Normal hyperplane sections of normal schemes in mixed characteristic

TL;DR

This paper proves the existence of infinitely many normal effective Cartier divisors on certain affine flat normal schemes over local Dedekind schemes in mixed characteristic, extending Bertini theorems and analyzing divisor class groups.

Contribution

It establishes the Bertini theorem for normal schemes in mixed characteristic and applies it to divisor class groups, advancing understanding in algebraic geometry.

Findings

Existence of infinitely many normal effective Cartier divisors under specified conditions

Extension of Bertini theorem to certain normal schemes in mixed characteristic

Results on the restriction map of divisor class groups 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.