On the family of affine threefolds \(x^my = F(x,z,t)\) (Q2877510)

The author after her path-breaking paper on failure of cancellation in positive characteristics [Invent. Math. 195, No. 1, 279--288 (2014; Zbl 1309.14050)] generalizes her results to answer a question raised by P. Russell. Consider the ring \(A=k[X,Y,Z,T]/(X^mY-F(X,Z,T))\) with \(m>1\) and assume it is an integral domain, where \(k\) is any field. Let \(f(Z,T)=F(0,Z,T)\). Part of the main theorem then states that \(A\) is isomorphic to a polynomial ring in three variables if and only if \(f(Z,T)\) is a variable in \(k[Z,T]\). These conditions are also equivalent to saying that \(X^mY-F(X,Z,T)\) is a variable in \(k[X,Y,Z,T]\), which answers the question of Russell. Using an example of \textit{T. Asanuma} [Invent. Math. 87, 101--127 (1987; Zbl 0607.13015)], the author had previously constructed examples of threefolds in positive characteristics which does not satisfy cancellation. The present result yields such families of counterexamples. The proof of the main theorem occasionally uses some basic \(K\)-theory.