54. Nikhilesh Dasgupta On the family of affine threefolds xmy F(x, z, t) II

In these lectures, we shall study the affine threefold V given by xmy F(x, z, t) for natural numbers m over any field k. We shall use the theory of exponential maps to prove that when m is at least 2, V is isomorphic to A3k if and only if f(z, t) : F(0, z, t) is a coordinate of kz, t. In particular, when char(k) p 0 and f defines a nontrivial line in the affine plane A2k (such a V will be called an Asanuma threefold), then V is not isomorphic to A3k although V A1k is isomorphic to A4k; thereby providing a family of counterexamples of the Zariski cancellation conjecture for the affine 3space in positive characteristic. These talks will be based on the paper of Neena Gupta 1 with the same title. References: 1 N. Gupta, On the family of affine threefolds xmy F(x, z, t), Compositio Math. 150 (2014), 979998.