On rigidity of Pham-Brieskorn surfaces

TL;DR

This paper extends the understanding of rigidity of Pham-Brieskorn surfaces over arbitrary fields, providing new conditions for rigidity and stable rigidity, and applying these results to the cancellation problem in algebraic geometry.

Contribution

It offers new sufficient conditions for rigidity of Pham-Brieskorn domains over fields of any characteristic, and applies these to the Zariski Cancellation Problem in characteristic p.

Findings

Established conditions for rigidity over arbitrary fields.

Proved rigidity of certain rings with added polynomial terms.

Connected rigidity results to the non-existence of non-trivial exponential maps.

Abstract

It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\geq 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:= k[X,Y,Z]/(X^a + Y^b + Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\leq 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\geq 0$ 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 $k[X,Y,Z,T]/(X^mY+T^{p^rq} + Z^{p^e})= k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\geq 1$, 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 $a,b,c\geq 2$ with $gcd(a,b,c) = 1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]/(X^aY^b + Z^c+ F(Y))$ is a rigid domain.