Uniform symbolic topologies and hypersurfaces (Q6563080)

Let \(P\) denote a prime ideal of Noetherian ring \(R\) and let \(P^{(t)} = P^tR_P \cap R\) denote the \(t\)-th symbolic power, \(t \in \mathbb{N}\). Let \(R\) be a normal, excellent local domain and \(X \subset \operatorname{Spec} R\). Suppose that there exits a positive integer \(b\) such that \(P^{(bn)} \subseteq P^n\) for all \(n \geq 1\) and all \(P \in X\). Then \(X\) is called to satisfy the uniform symbolic topology property on prime ideals. In the past there are several investigations when \(P^{(bn)} \subseteq P^n\) for all \(n \geq 1\) for some particular prime ideals. Here it is shown that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. Moreover, the authors provide bootstrapping techniques and apply them to give families of prime ideals in hypersurfaces of positive characteristic which have uniform symbolic topologies.