Self-intersection formula for a toric surface divisor
= 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.