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Publication

Frobenius techniques in birational geometry

Zsolt Patakfalvi
2017
Conference Papers

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.

Official source

Zsolt Patakfalvi
2017
Conference Papers

Abstract

Official source

Related concepts (19)

Algebraic geometry

Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations.

Complex geometry

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

Birational geometry

In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. A rational map from one variety (understood to be irreducible) to another variety , written as a dashed arrow X Y, is defined as a morphism from a nonempty open subset to .

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