Fine behavior of symbolic powers of ideals

Abstract

A fundamental property connecting the symbolic powers and the usual powers of ideals in regular rings was discovered by Ein, Lazarsfeld, and Smith in 2001, and later extended by Hochster and Huneke in 2002. In this paper we give further generalizations which give better results in case the quotient of the regular ring by the ideal is F-pure or F-pure type. Our methods also give insight into a conjecture of Eisenbud and Mazur concerning the existence of evolutions. The methods used come from tight closure and reduction to positive characteristic.