Self-intersection formula for a toric surface divisor

In a smooth complete toric surface, if consecutive primitive rays are $u,w,v$ and $u+v=aw$, then the invariant divisor $D_w$ has $D_w^2=-a$. A ray introduced inside a cone by a smooth subdivision has $a>0$, so its divisor has negative self-intersection.

A compact invariant curve with consecutive rays $u,v,w$ on a smooth toric surface contracts torically to a smooth point exactly when $u+w=v$, equivalently when its self-intersection is $-1$. Deleting $v$ then reverses a toric blowup at a torus-fixed point.

// Target: geometry-and-topology.bigb