The generalized stable equivalence problem

In this note we study the following problem. Let k be an algebraically closed field and X be an affine variety over k. Suppose that H-1, H-2 subset of X are two hypersurfaces such that there exists an automorphism f of X x k(n) satisfying f (H-1 x k(n)) = H-2 X k(n) for some n > 0. Does this imply that there exists an automorphism (f) over tilde of X such that (f) over tilde (H-1) = H-2? We give an affirmative solution if one hypersurface is not k-uniruled and a counterexample in general. (C) 2012 Elsevier Inc. All rights reserved.