Why take the dual cone when constructing toric variety?

Question

I am reading an introduction to toric varieties. I don't understand why we are taking the monoid associated to the dual cone instead of simply taking the monoid associated to the cone. Is there any conceptual/pratical reason to do this ?

Answer 1

I think the construction of toric varieties from cones might seem a bit confusing and unmotivated at first, so you can go the other way: define what a toric variety is and then recover the underlying cones.

So a toric variety is an irreducible variety $X$ containing a torus $(\mathbb{C}^\times)^n$ as a Zariski open subset such that the action of the torus on itself extends to its action on $X$. (This is why a toric variety is called toric!)

Then one can show that every normal separated toric variety comes from a fan $\Sigma$, which is a collection of strongly convex rational polyhedral cones that "fit nicely" together (if $\sigma \in \Sigma$, then all faces of $\sigma$ lie in $\Sigma$, and all cones from $\Sigma$ intersect by their faces; in particular, intersections $\sigma\cap\tau$ lie in $\Sigma$ as well). From a fan $\Sigma$ you can glue a toric variety $X_\Sigma$, and for any normal separated $X$ there is a fan $\Sigma$ such that $X \cong X_\Sigma$. Cones of your fan give the affine pieces of $X_\Sigma$, and to have this nice correspondence between toric varieties and fans, you really need to pass to the dual cones.

This is explained in chapter 3 of the book by Cox, Little, and Schenck.