On algebraic varieties over fields of prime characteristic (Q1842022)

Let \(K\) be a field of characteristic \(p\), \(L\) be an algebraic function field over \(K\) and let \(X\) be a reduced irreducible algebraic variety over \(K\) whose rational functions field is \(L\). Then there is a canonical sequence of subfields \(L = L_ 0 \supset L_ 1 \supset L_ 2 \supset \cdots\) with \(L_ i = K(L^{p^ i})\), \((i \in \mathbb{N})\) and an induced sequence of finite purely inseparable morphisms of algebraic varieties \[ X = X_ 0 \to X_ 1 \to X_ 2 \to \cdots\;. \] This note is devoted to the study of the properties of \(X_ i\), with special regard to geometric normality and smoothness.