On the family of affine threefolds x^m y = F(x, z, t) - II (Nikhilesh Dasgupta)
The talk of Nikhilesh Dasgupta at the Seminar of the Laboratory on Algebraic Transformation Groups.
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 A3 k if and only if f(z, t) := F(0, z, t) is a coordinate of k[z, t]. In particular, when char(k) = p > 0 and f defines a non-trivial line in the affine plane A2 k (such a V will be called an Asanuma threefold), then V is not isomorphic to A3 k although V × A1 k is isomorphic to A4 k; thereby providing a family of counter-examples of the Zariski cancellation conjecture for the affine 3-space in positive characteristic. These talks will be based on the paper of Neena Gupta  with the same title.
 N. Gupta, On the family of affine threefolds xmy = F(x, z, t), Compositio Math. 150 (2014), 979-998.