= Four-ray completion criterion for a cyclic quotient surface singularity
{title2=$s\mid r-1,\quad s\equiv\pm q\pmod r$}
The <cyclic quotient surface singularity> $\frac1r(q,1)$ is the only singular point of a complete four-ray <toric surface> exactly under the displayed condition. Normalize the three smooth cones to the consecutive rays $(a,-1),(1,0),(0,1),(-1,b)$. The singular cone has determinant $1-ab=r$, so $ab=1-r$. In the basis $(-1,b),(0,-1)$, its other generator is $-a(-1,b)+r(0,-1)$, giving quotient parameter $a\bmod r$. Thus $a=\pm s$ for a positive divisor $s$ of $r-1$. Conversely these rays construct the surface. Exchanging the two axes replaces $q$ by $q^{-1}$; replacing a factor by its complementary factor preserves the criterion under this equivalence.
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