A GAGA type theorem on germs of analytic curves through germs of quotient surface singularities (Q1322608)

The author proves that the germ of a reduced complex analytic curve \((C,x)\) through a quotient singularity \((X,x)\) of dimension two is biholomorphic to the germ \((Y,D,y)\) of a projective curve on a projective surface. Along the way he gives an explicit description of the divisor class group of a quotient singularity. A detailed, self-contained proof is given; the key point is that equivariant functions on \((\mathbb{C}^ 2,0)\) with isolated singularity are finitely determined. The proper context, especially for generalisations, is the theory of ``geometrically defined subgroups of \({\mathcal A}\) and \({\mathcal K}\)'' [cf. \textit{J. Damon}, Proc. Symp. Pure Math. 40, Part 1, 233- 254 (1983; Zbl 0519.58014)].