Projective equivalence of plane curve singularities defined by the homogenization of weighted homogeneous polynomials in \(\mathbb{C}[Y,Z]\) and its difference from their analytic equivalence (Q1018240)

The author considers the following three type of equivalences: local analytic, local topological and global projective. The objects are isolated singularities described by weighted homogeneous polynomials in two variables and projective plane curves associated to these polynomials. Let \(f(y,z)\) be such a weighted homogeneous polynomial with an isolated singularity at \(0\) (it implies that \(f\) has no multiple factors in \(\mathbb{C}[y,z])\) and \(F(x,y,z)\) be its homogenization. The author studies, in an elementary way, the problem of differences between analytical equivalence of such singularities \(f=0\) and \(g=0\) and projective equivalence of \(F(x,y,z)=0\) and \(G(x,y,z)=0\) in \(\mathbb{P}^{2}\mathbb{C}\). The paper is very long with many unnecessary repetitions in formulations of facts and proofs.