On rigidity of Pham-Brieskorn surfaces

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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 1a+1b+1c≤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∤mq and e>r≥1, fixing y, a crucial result used in [10] to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine 3-space in positive characteristic.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.