The biregular cancellation problem

Abstract

The main purpose of this paper is to study the relationship of the above two problems. We apply the results of the birational cancellation problem [17, 4 and 13] to investigate the biregular cancellation problem. Theorem 5 is a general form of [1, (7.5) Corollary and (7.6) Corollary] and partially answers a question of [1]. Theorem 7 is the 1-dimensional biregular cancellation which was proved by Abhyankar, Eakin and Heinzer [1]; subsequently Miyanishi gave another proof from the viewpoint of Ga-action [14]. Theorems 14, 15 and 16 deal with the biregular cancelation problem of nonrational surfaces. Theorem 16 is implicit in the works of Iitaka and Fujita [7 and 9]. I would like to thank Prof. T. Sugie, Kyoto University, who called my attention to Iitaka's theory of quasi-Albanese maps [7]. Theorem 6 and its corollary are the non-algebraic version of the absence of nontrivial forms of A2; they answered a question of Prof. W. HeinTer in a private discussion. Theorem 13 generalizes (4.1) Theorem of [1].