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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-152/1/a/iii/solution