A principal minor of a matrix is one whose row and column indices are the same. It’s a theorem that the ideal generated by the size two principal minors of a square matrix of variables in the polynomial ring over those variables is toric. A toric variety is an irreducible algebraic set that contains an algebraic torus that is Zariski open, such that the action of the torus on itself extends to an action on the variety. Toric varieties are an active area of research, and have applications in mirror symmetry, coding theory, algebraic statistics, and geometric modeling! In this project, we will investigate some of the properties of the principal 2-minor ideals by studying their corresponding algebraic sets as a toric varieties.
Here are some of the questions we will try to answer:
- What is the rational polyhedral cone that gives the defining semigroup for the coordinate ring of the variety? Is it regular? Is it simplicial? What is its dual?
- Provide another proof principal 2-minor ideals are normal, using toric geometry.
- Is the variety compact?
- What is the Weil divisor class group for the variety? What is the Picard group?
- And more!
It is recommended that you have access to copies of Cox, Little, & Schenck’s Toric Varieties and Sturmfels’s Groebner bases and convex polytopes.