Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms
Let \(V_{s,t}\) be the germ at the origin of the curve defined by \[ f_{s,t}:=x^5+sx^4y+tx^3y^2+y^5=0\;,\;s,t\in\mathbb{C}\;. \] The biholomorphically equivalence problem is discussed for the family \(V_{s,t}\). It is proved that \(V_{s_1, t_1}\) and \(V_{s_2, t_2}\) are biholomorphically equivalent if and only if \(j(s_1, t_1)=j(s_2, t_2)\), \(k(s_1, t_1)=k(s_2, t_2)\) and \(l(s_1, t_1)=l(s_2, t_2)\). Here \(j, k, l\) are relatively complicated invariants from classical invariant theory, for example \[ j(s,t)=\frac{5(1440000\cdot 10!)^2(125-3st^2)^2}{256s^5-1600s^3t-27s^2t^4+2250st^2+108t^5+3125}. \]
scientific article; zbMATH DE number 6353228
scientific article