On the rigidity of certain Pham-Brieskorn rings

Abstract

Fix a field k of characteristic zero. If a1,…,an (n≥3) are positive integers, the integral domainBa1,…,an=k[X1,…,Xn]/〈X1a1+⋯+Xnan〉 is called a Pham-Brieskorn ring. It is conjectured that if ai≥2 for all i and ai=2 for at most one i, then Ba1,…,an is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D:B→B is the zero derivation.) The conjecture is known to be true when n=3, and in certain special cases when n≥4. This article settles several cases not covered by previous results. For instance, we show that if a≥n≥4 then Ba,…,a is rigid (where ‘a’ occurs n times), and that if ∑i=1n1ai≤1n−2 then Ba1,…,an is stably rigid.