Purely inseparable coverings of exponent one of the affine plane (Q1068141)

Let k be an algebraically closed field of characteristic p\(>0\). A purely inseparable covering of exponent 1 of the affine plane \({\mathbb{A}}^ 2:=Spec k[u,v]\), or affine Zariski surface, is a normal affine surface \(X:=Spec {\mathbb{A}}\) such that \(k[u,v]\subsetneqq A\subsetneqq k[x,y]\) with \(u=x^ p,\) \(v=y^ p.\) \textit{J. Lang} [Mich. Math. J. 28, 375-380 (1981; Zbl 0495.14021)] has shown that if \(char(k)=2\), then \(A:=k[u,v(uv+1)x+vy]\) is a regular, factorial, irrational k-algebra, and we will show that this is true, in fact, for arbitrary \(p>0\). These examples suggest that affine Zariski surfaces have a rich geometry to be studied. One should contrast this with the situation when \(A\subset k[x,y]\) is an affine, regular k-algebra such that k[x,y] is generically separable over A. It is known that then A is a polynomial ring over k if A is factorial.