On pencils of polynomial curves (Q1921356)

Let \(k\) be a field. An irreducible \(F \in k[X,Y]\) is called a polynomial curve if the plane curve that it determines is rational and has one rational place at infinity. In the characteristic 0 case, it is known that those \(F\) which satisfy: \[ F-\lambda\text{ is a polynomial curve for infinitely many }\lambda \in k\tag{*} \] are precisely the variables of \(k[X,Y]\), i.e. \(k[X,Y] = k[F,G]\) for some \(G\). This paper gives an analogous characterization in the positive characteristic case. The characteristic 0 Bertini theorem yields that if \(F\) is generally a polynomial curve then \(F - \lambda\) is a nonsingular polynomial curve (for general \(\lambda)\), hence a line, and so \(F\) is a variable by the epimorphism theorem. Let \(k\) have characteristic \(p>0\) and let \(\overline k\) denote the algebraic closure of \(k\). Let \(R = k^{[2]}\), the polynomial algebra in two variables over \(k\). An element \(u\) of \(R\) is called a \(p\)-generator of \(R\) if there exists \(v\in R\) and \(n\geq 0\) such that \(k[u,v] \supseteq R^{p^n}\). Let \(\tau\) be an indeterminate over \(R\). Then the author proves the equivalence of the following properties: (1) \(u\) is a \(p\)-generator of \(R\) which does not belong to \(k [R^{p^n}]\); (2) \(u-\lambda \in \overline k \otimes R\) is a polynomial curve for all \(\lambda \in \overline k\); (3) \(u-\lambda \in \overline k \otimes R\) is a polynomial curve for infinitely many \(\lambda \in \overline k\); (4) \(u-\tau \in \overline{k(\tau)} \otimes R\) is a polynomial curve; (5) \(u-\tau \in \overline k(\tau)^{p^{-\infty}} \otimes R\) is a polynomial curve; (6) \(u-\tau \in k(\tau)^{p^{-\infty}} \otimes R\) is a polynomial curve.