Using the notion of uniruledness we indicate a class of algebraic varieties which have a stronger version of the cancellation property. Moreover, we give an affirmative solution to the stable equivalence problem for non-uniruled hypersurfaces.

We study the following cancellation problem over an algebraically closed field of characteristic zero. Let X, Y be affine varieties such that $X{\times }^m\cong Y{\times }^m$ for some m. Assume that X is non-uniruled at infinity. Does it follow that X ≅ Y? We prove a result implying the affirmative answer in case X is either unirational or an algebraic line bundle. However, the general answer is negative and we give as a counterexample some affine surfaces.

Using BMY inequality and a Milnor number bound we prove that any algebraic annulus ${ℂ}^{*}$ in ${ℂ}^2$ with no self-intersections can have at most three cuspidal singularities.