Ibadan Lectures on Toric Varieties

TL;DR

This paper provides an accessible overview of toric varieties, highlighting their combinatorial structure, mathematical significance, and diverse applications across various fields of mathematics and science.

Contribution

It significantly extends Frank Sottile's original lectures, offering a comprehensive and detailed exposition on the theory and applications of toric varieties.

Findings

Toric varieties can be fully understood through geometric combinatorics.

They are widely applicable in areas like tensors, statistical models, and geometric modeling.

The notes expand on existing lectures, providing deeper insights into the subject.

Abstract

Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely understood through objects from geometric combinatorics. While accessible and understandable, the class of toric varieties is also rich enough to illustrate many properties of algebraic varieties. Toric varieties are also ubiquitous in applications of mathematics, from tensors to statistical models to geometric modeling to solving systems of equations, and they are important to other branches of mathematics such as geometric combinatorics and tropical geometry. These notes are based on, and significantly extend, Frank Sottile's short course of four lectures at the CIMPA school on Combinatorial and Computational Algebraic Geometry in Ibadan, Nigeria 12--23 June 2017.