Toric Geometry Examples

In the words of Prof. Melody Chan, toric geometry is interesting because it yields examples entirely controlled by combinatorics; we can associate toric varieties to polyhedral fans by gluing together the prime spectra of polynomial rings given by adjoing semigroups associated to those fans to a ground field (usually $\mb C$). Rarely is such robust gluing data given in such a conscise and accessable manner.

• Toric Surfaces from Fans

  Here are examples of 2-dimensional toric varieties with all the affine spectra worked out and the gluing data specified explicitly to the degree necessary to work out e.g. Čech cohomology by hand. I’ve used sage at times, and will include the code I’ve utilized. I’ll also include some visualizations of the cones when possible. I use $\hull(\{v_i\})$ to denote the convex hull of the $v_i$, and $\cone(\{v_i\})$ to denote the set $\{a_1v_1+a_2v_2+\cdots\ \text{s.t.}\ a_i\in \mb R_{\geq 0}\}$. If $\sigma$ is a cone, $\sigma^\nu = \{v\in\mathbb R^n \text{ s.t. } v\cdot w\geq 0\ \ \forall w\in \sigma\}$ is the dual of $\sigma$.