##### Department of Mathematics,

University of California San Diego

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### Southern California Algebraic Geometry Seminar

## Karl Schwede

#### Penn State

## F-singularities in families

##### Abstract:

F-singularities are classes of singularities defined by the behavior of Frobenius. A prominent tool for measuing these singularities is the test ideal, a characteristic $p > 0$ analog of the multiplier ideal. Recently, there has been interest in applying the methods of F-singularities to a number of geometric problems in positive characteristic. However, one gap in the theory has been the behavior of F-singularities in families. For example generic restriction theorems for test ideals have been lacking. In this talk, I will discuss recent joint work with Zsolt Patakfalvi and Wenliang Zhang on the behavior of F-singularities and test ideals in families. I will first define the relevant terms and explain why you these definitions and methods are useful (as a replacement for Kodaira vanishing in characterisic $p > 0$). We will then obtain generic (and non-generic) restrictions theorems for test ideals. Some global geometric consequences will also be discussed if there is sufficient time.

Dragos Oprea

### April 13, 2013

### 2:00 PM

### AP&M 6402

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