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. Suppose K(t) k(t1,t2,t3), then K k(t1,t2). 2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let A=K[x,y,z,w]/M be the coordinate ring of M. Suppose A[t] k[x1,x2,x3,x4], then frac(A) k(x1,x2,x3), where frac(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].