Affine surfaces with isomorphic \(\mathbb{A}^2\)-cylinders (Q2312762)

The generalized Zariski cancellation problem asks whether two smooth affine varieties \(X,Y\) are isomorphic if \(X\times\mathbb{A}^n\) is isomorphic to \(Y\times\mathbb{A}^n\) for some \(n\). In positive characteristics, even for \(X=\mathbb{A}^3\), \textit{N. Gupta} [Invent. Math. 195, No. 1, 279--288 (2014; Zbl 1309.14050)] showed that the answer is in the negative. In characteristic zero, for curves an affirmative answer was given by \textit{S. S. Abhyankar} et al. [J. Algebra 23, 310--342 (1972; Zbl 0255.13008)] and for \(X=\mathbb{A}^2\), this was settled affirmatively by \textit{T. Fujita} [Proc. Japan Acad., Ser. A 55, 106--110 (1979; Zbl 0444.14026)]. There are well known counter examples to the problem with \(n=1\) and \(\dim X\geq 2\) due to \textit{M. Hochster} [Proc. Am. Math. Soc. 34, 81--82 (1972; Zbl 0233.13012)] over the reals and an unpublished work of W. Danielewski over the complex numbers [\textit{G. Freudenburg}, Algebraic theory of locally nilpotent derivations. Berlin: Springer (2006; Zbl 1121.13002), Chapter 10]. In this paper, the author studies the case of rational surfaces \(X,Y\) with \(X\times \mathbb{A}^2\cong Y\times\mathbb{A}^2\), but \(X\times\mathbb{A}^1\) is not isomorphic to \(Y\times\mathbb{A}^1\). The main theorem says the following: Let \((V_i,C_i)\), \(i=1,2\) be pairs consisting of a smooth cubic surface \(V_i\subset\mathbb{P}^3\) and a cuspidal hyperplane section \(C_i\) of \(V_i\). Let \(X_i-V_i-C_i\). Then, \(X_1\times\mathbb{A}^2\cong X_2\times\mathbb{A}^2\), whereas, if \(X_1\times\mathbb{A}^1\cong X_2\times \mathbb{A}^1\), then \((V_1,C_1)\cong (V_2, C_2)\).