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Ibadan Lectures on Toric Varieties

Frank Sottile

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.

Ibadan Lectures on Toric Varieties

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.

Paper Structure

This paper contains 13 sections, 23 theorems, 104 equations, 11 figures.

Table of Contents

  1. Affine Toric Varieties
  2. Affine Toric Varieties
  3. Toric Ideals
  4. Toric Projective Varieties and Solving Equations
  5. Toric Varieties in Projective Space
  6. Kushnirenko's Theorem
  7. Toric Varieties From Fans
  8. Cones and Fans
  9. Toric Varieties From Fans
  10. The Double Pillow
  11. Bernstein's Theorem and Mixed Volumes
  12. Mixed Volumes
  13. Bernstein's Theorem

Key Result

Theorem 1.2

The toric ideal $I_{\mathcal{A}}$ is a prime ideal. As a complex vector space, it is spanned by the binomials Eq:toric_binomials.

Figures (11)

  • Figure 1: Exponents ${\mathcal{A}}$ and the twisted cubic.
  • Figure 2: The hexagon and its lift.
  • Figure 3: Curves of polynomial system \ref{['Eq3:ex_sparse']}.
  • Figure 4: Linear section of cubic surface.
  • Figure 5: Monoid generated by the lift of $\{0,2,3\}$.
  • ...and 6 more figures

Theorems & Definitions (50)

  • Example 1.1
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 2.1
  • Example 2.2
  • Example 2.3
  • ...and 40 more