  In recent years, a multiplier ideal defined on arbitrary varieties, so called
Mather multiplier ideal, has been developed independently by Ein-Ishii-Mustata,
and de Fernex-Docampo. With this new tool, we have a chance of extending some
classical results proved in nonsingular case to arbitrary varieties to
establish their general forms. In this paper, we first extend a result of
geometric nullstellensatz due to Ein-Lazarsfeld in nonsingular case to any
projective varieties. Then we prove a result on comparison of symbolic powers
with ordinary powers on any varieties, which extends results of
Ein-Lazarsfeld-Smith and Hochster-Huneke.
