Christian Haase, FU-Berlin Two Constructions for Quadratic Gr\"obner Bases

Toric ideals are combinatorial objects which are used in commutative algebra and algebraic geometry as a test ground for general theories. Beyond their use as a source of examples, many general theorems can be reduced to the toric case via degenerations.

In this talk, I will stay on the combinatorial side of things. In the first half, I will explain how we can "see" toric ideals and their Gr\"obner bases in convex geometry, and I will translate a conjecture which, in algebraic geometry language, states that the defining ideal of a smooth projective toric variety has a quadratic Gr\"obner basis.

In the second half, I will present two constructions for such Gr\"obner bases: hyperplane subdivisions, and project-and-lift. I will illustrate their use on $3\times 3$ transportation polytopes.

This is joint work with Andreas Paffenholz.