A generalisation of Noether's formula for the number of virtual double points to space curve singularities (Q584336)

By a well-known theorem of Max Noether the \(\delta\)-invariant of a plane curve singularity is equal to \(\sum 12n_ P(n_ P-1) \), where the sum ranges over all infinitely near points. One can view the terms in this sum as dimensions of certain cohomology groups on the exceptional divisor in the embedded resolution. This interpretation is generalised in this note to space curves \((X,0)\subset ({\mathbb{C}}^ n,0)\). The resulting formula involves the superabundance of k-tics through the intersection of the strict transform of X and the exceptional divisor E of the blow-up of \({\mathbb{C}}^ n\), but it also takes embedded components of the tangent cone of X into account. An example is given of a curve where the number of double points in an actual deformation is strictly less than the number of virtual double points.