Fiber cones and the integral closure of ideals

Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. This paper studies the question of when mI is integrally closed. Particular attention is focused on the case R is a regular local ring and I is a reduced ideal. This question arose through a question posed by Eisenbud and Mazur on the existence of evolutions. Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. In this paper we are interested in the question of when mI is integrally closed. To explain our motivation for studying this question, we recall the definition of an evolution [5]. Definition 1.1. Let k be a ring and let T be a local k-algebra essentially of finite type over k. An evolution of T over k is a local k-algebra R, essentially of finite type, and a surjection R → T of k-algebras inducing an isomorphism ΩR/k ⊗R T ∼= ΩT/k. The evolution is trivial if R → T is an isomorphism, and T is evolutionarily stable over k if all evolutions are trivial. It is possible that over a field k of characteristic 0, every reduced local k-algebra T essentially of finite type over k is evolutionarily stable. No counterexamples are known. See [5, 7, 13] for some partial results. The question concerning the existence