Frobenius techniques in birational geometry

TL;DR

This survey reviews the recent use of Frobenius techniques in positive characteristic algebraic geometry, explaining fundamental ideas and summarizing key results achieved with these methods.

Contribution

It provides a comprehensive overview of how Frobenius methods have been applied to obtain new results in algebraic geometry in positive characteristic.

Findings

Summarizes recent algebraic geometry results using Frobenius techniques

Explains fundamental ideas behind Frobenius methods

Highlights the impact of Frobenius techniques in positive characteristic geometry

Abstract

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.