Invariants of singular points of algebraic curves (Q800439)

Let \(F\subset {\mathbb{C}}P^ 2\) be a plane curve and \(p\in F\) a singular point of F. The author establishes the following equality of local numerical invariants of the singularity (F,p): \(h(p)=2{\mathcal H}(p)+2g(p)+s^*(p),\) where \({\mathcal H}(p)\) is the intersection number p of the curve F and a generic curve whose equation is of the form: \(\sum^{2}_{i=0}q_ i(\partial F/\partial Z_ i)=Q,\) g(p) is the so- called genus of (F,p) defined by using \({\mathcal H}(p)\) and the degrees of the branches of F passing through p, \(s^*(p)=\sum_{P}(\alpha (P)-1)\) (where the sum is taken over all branches P of F passing through p), and h(p) is the intersection number at p of F and its Hessian. This formula is valid under the additional hypothesis that F has no multiple and linear components passing through p, and in this context was conjectured by \textit{D. A. Gudkov} [Russ. Math. Surv. 29, No.4, 1-79 (1974), translation from Usp. Mat. Nauk 29, No.4, 3-79 (1974; Zbl 0316.14018)].