2023/10/03 by Neena Gupta, Gupta, Neena, Ananya Pal +1
Mathematics · #13A02 (Secondary) #14R20 (Primary) 13A50 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2310.01864
openalex publication_date 2023/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that, over an algebraically closed field k of characteristic zero, for any three integers a,b,c≥ 2, any Pham-Brieskorn surface B(a,b,c):= k[X,Y,Z]/(Xa + Yb + Zc) is rigid when at most one of a,b,c is 2 and stably rigid when (1)/(a) + (1)/(b) + (1)/(c)≤ 1. In this paper we consider Pham-Brieskorn domains over an arbitrary field k of characteristic p≥ 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]/(XmY+Tprq + Zpe)= k[x,y,z,t], for m,q>1, p\nmid mq and e>r≥ 1, fixing y, a crucial result used in the paper "On the cancellation problem for the affine space \mathbbA3 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≥ 2 with gcd(a,b,c) = 1 and for F(Y)∈ k[Y], the ring k[X,Y,Z]/(XaYb + Zc+ F(Y)) is a rigid domain.