Algebraic Geometry Seminar: Kevin Tucker (University of Utah)

After reviewing some basic constructions with multiplier ideals on complex algebraic varieties, we recall the definition of multiplier ideals in positive characteristic and highlight the failure of some desirable properties to carry over in this setting. This leads us to a related measure of singularities coming from commutative algebra -- the test ideal -- which seems to exhibit better behavior than the multiplier ideal in positive characteristic. While test ideals were first introduced in the theory of tight closure, our goal in this talk will be to describe a new and algebro-geometric characterization of test ideals using regular alterations. This characterization is also holds for multiplier ideals in characteristic zero (but not in positive characteristic!!!), providing a kind of uniform description with new insight and intuition. Time permitting, we will use this result to give an analogue of Nadel Vanishing in positive characteristic.