Abstract:
In these lectures, we shall study the affine threefold $V$ given by $x^my = F(x, z, t)$ for natural numbers $m$ over any field $k$. We shall use the theory of exponential maps to prove that when $m \geqslant 2$, $V$ is isomorphic to $\mathbb{A}^3_k$ if and only if $f(z, t) := F(0, z, t)$ is a coordinate of $k[z, t]$. In particular, when $\mathrm{char}(k) = p > 0$ and $f$ defines a non-trivial line in the affine plane $\mathbb{A}^2_k$ (such a $V$ will be called an Asanuma threefold), then $V$ is not isomorphic to $\mathbb{A}^3_k$ although $V \times \mathbb{A}^1_k$ is isomorphic to $\mathbb{A}^4_k$; 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 [1] with the same title.
References:
[1] N. Gupta, On the family of affine threefolds $x^my = F(x, z, t)$, Compositio Math. 150 (2014), 979-998.
