On rigidity of Pham-Brieskorn surfaces

Journal of Algebra. 2024. Vol. 657. P. 748–772.

Gupta N., Pal A.

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]/(X^a + Y^b + Z^c ) 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 ]/(X^mY + T^{p^{r}q} + Z^{p^e} ) = k[x, y, z, t], for m, q > 1, e > r ≥ 1 and p not dividing mq; fixing y, a
crucial result used in ``On the cancellation problem for the affine space A^3 in characteristic p, 
Invent. Math. 195" by Neena Gupta to show that the Zariski Cancellation Problem 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]/(X^aY^b + Z^c + F (Y)) is a rigid domain.

Keywords: Makar-Limanov invariant, инвариант Макар-Лиманова, Exponential Map, экспоненциальное отображение, Pham-Brieskorn surface, rigid domain, stably rigid domain, поверхность Фама-Брискорна, жёсткая область целостности, стабильно жёсткая область целостности