On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) K₁ and K₂ over a field k, a k-isomorphism of K₁ [t] (K₁ (t)) and K₂ [t] (K₂ (t)) implies a k-isomorphism of K₁ and K₂. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:. 1. Let K be an affine field over an algebraically closed field k of any characteristic. SupposeK (t) ≃ k (t₁, t₂, t₃), thenK ≃ k (t₁, t₂) . 2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. LetA = K [x, y, z, w] / M be the coordinate ring of M. SupposeA [t] ≃ k [x₁, x₂, x₃, x₄], thenfrac (A) ≃ k (x₁, x₂, x₃), wherefrac (A) is the field of fractions of A. In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171].