On the number of branches for real and complex weighted homogeneous curve singularities (Q757657)

In this paper formulae for the number of branches for real and complex weighted homogeneous curve singularities are established in terms of the Jacobian algebra of the curve singularity. Let \((X,0)\subset ({\mathbb{C}}^ m,0)\) be a complex weighted homogeneous curve singularity, which is Gorenstein and reduced, then the number of complex branches equals \(\dim_{{\mathbb{C}}}(J(X)_ s)+1,\) where J(X) is the Jacobian algebra having the middle weight s such that 2s is the maximal nonzero weight in J(X), and \(J(X)_ s\) is its subspace generated by elements of weight s. As for the real case \((X,0)\subset ({\mathbb{R}}^ m,0)\), the number of branches equals \(sign(J(X)_ s)+1,\) where \(sig(J(X)_ s)\) denotes the signature of the nonsingular pairing: J(X)\(\times J(X)\to R\) restricted on \(J(X)_ s\). In both cases, the Jacobian algebras have better properties, the formulae turn to be much simpler than the general case.