Criteria for the canonicity of cyclic quotients of regular and nondegenerate double singular points (Q1095978)

Let k be an algebraically closed field with \(char(k)=0\), \(\mu_ r\) the cyclic group of \(r^{th}\) roots of unity, \(a_ 1, a_ 2, a_ 3\) integers such that \(0<a_ i<r\), \((a_ i,r)=1\). Then the factor variety \(A^ 3/\mu_ r\) given by the action \((x_ 1,x_ 2,x_ 3)\to (\epsilon^{a_ 1}x_ 1,\epsilon^{a_ 2}x_ 2,\epsilon^{a_ 3}x_ 3)\), \(\epsilon \in \mu_ r\) is a canonical singularity iff for every \(t\in {\mathbb{Z}}\), \(1\leq t\leq r-1\), it holds \(\{ta_ 1/r\}+\{ta_ 2/r\}+\{ta_ 3/r\}\geq 1\), where \(\{\) \(b\}\) denotes the fractional part of a rational number b. Finally the author studies canonical factors of quadratic singularities.