Insert the ray generated by $w=(0,1)$ between $v_1=(1,0)$ and $v_2=(-1,3)$. Since
$$
\det(v_1,w)=1,
\qquad
\det(w,v_2)=1,
$$
the subdivided fan is smooth and gives a <toric resolution of singularities>. The unique exceptional divisor is $D_w$. As
$$
v_1+v_2=(0,3)=3w,
$$
the <self-intersection formula for a toric surface divisor> gives $D_w^2=-3$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 1 / a / iii / Solution

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.


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