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Texas A&M University
Mathematics

Geometry Seminar

Date: February 6, 2019

Time: 2:00PM - 3:00PM

Location: BLOC 220

Speaker: Jose Burgos Gil, ICMAT, Madrid

  

Title: Arithmetic of Toric Varieties, Lecture 1.

Abstract: Abstract: Toric varieties form a very rich family of algebraic varieties that provide examples where explicit computations can be made. There is a toric dictionary that translates algebro-geometric concepts to combinatorial concepts. With this dictionary many algebro-geometric quantities can be computed. For example the degree of an ample line bundle on a toric variety is essentially given by the volume of an associated convex polytope. In joint work with P. Philippon and M. Sombra we have extended the toric dictionary to relate arithmetic properties with convex analytical properties. For example the height of a toric variety with respect to a positive metrized line bundle can be computed as the integral of a convex function on the associated polytope.

The minicourse will consist of three lectures:

Lecture 1: Overview of the theory of toric varieties.
Lecture 2: The theory of heights and the analogy between geometry and arithmetic.
Lecture 3: Arithmetic properties of toric varieties.

Most of the material of the course is in the book:

Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque No. 360 (2014).