The cancellation problems asks whether two varieties \(X,Y\) are isomorphic if they are stably isomorphic meaning \(X\times \mathbb{A}^n\cong Y\times\mathbb{A}^n\). There are a small number of affirmative answers and a plethora of negative answers including a beautiful one by the first author [Invent. Math. 195, No. 1, 279–288 (2014; Zbl 1309.14050)] in positive characteristics and \(X=\mathbb{A}^3\). In dimension one, the answer is positive, and it is also known when \(X=\mathbb{A}^2\) [T. Fujita, Proc. Japan Acad., Ser. A 55, 106–110 (1979; Zbl 0444.14026)]. One of the counterexamples in dimension two were constructed by Danielewski [G. Freudenburg, Algebraic theory of locally nilpotent derivations. 2nd enlarged edition. Berlin: Springer (2017; Zbl 1391.13001)]. Denote by \(S_n\), the surface define by \(x^ny-P(z)=0\), where \(P\) has degree at least two and distinct roots, over complex numbers. Then \(S_n\not\cong S_m\) if \(m\neq n\), but they are stably isomorphic. These Danilewski surfaces have been studied extensively.
In the present article, the authors define what they call double Danielwski surfaces \(S_{d,e}\), which are complete intersections in 4-space defined by similar equations as above depending on two parameters \(d,n\) of natural numbers. They study these in depth and in particular, show that these are not isomorphic to to the original \(S_n\) and they too are stably isomorphic for a fixed \(d\), but \(S_{d,e}\not\cong S_{d,e+1}\), thus creating a new rich class of counterexamples to the cancellation problem.