On rigidity of Pham-Brieskorn surfaces

It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\ge 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:=k[X,Y,Z]\mathrm{/}(X^a+Y^b+Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\ge 0$ and give sufficient conditions on $(a,b,c)$ for which any Pham-Brieskorn domain $B_{(a,b,c)}$ is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on $k[X,Y,Z,T]\mathrm{/}(X^{m}Y+T^{p^{r}q}+Z^{p^e})=k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\ge 1$, fixing $y$, a crucial result used in the paper “On the cancellation problem for the affine space ${\mathbb{A}}^3$ in characteristic $p$” by first author, to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine $3$-space. We also provide a sufficient condition for $B_{(a,b,c)}$ to be stably rigid. Along the way we prove that for integers $a,b,c\ge 2$ with $gcd(a,b,c)=1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]\mathrm{/}(X^a{Y}^b+Z^c+F(Y))$ is a rigid domain.