= Two minimal toric resolutions of a four-ray threefold cone
{title2=$a=(1,0,0),\ b=(1,1,1),\ c=(0,2,1),\ d=(1,1,0)$}
For this <toric cone>, one smooth <fan subdivision> has maximal cones $abc,acd$ and no exceptional divisors. Another has $abd,bce,cde,dbe$, with $e=a+c=(b+c+d)/2$. Its only exceptional divisor is a <projective plane>, since the quotient fan at $e$ has the three rays $\bar b,\bar c,\bar d$ with sum zero and unimodular neighboring cones. Both subdivisions admit no proper smooth coarsening. Their common smooth refinement is obtained from the first by adding $e$ in $ac$ and then $w=a+e$ in $ae$, and from the second by adding $w=b+d$ in $bd$. Each step is a <toric blowup along an orbit closure>.
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