Uniform Symbolic Topologies and Hypersurfaces

We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show 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. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.