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

Abstract

A FORMULA for the number of real branches for a curve singularity defined by a smooth germI: R”+‘, 0 -+ R”, 0 was discovered by Aoki. Fukuda. and Nishimura [Z]. They showed that this number can be computed as the degree of an associated mapping. This degree can, in turn, be computed algebraically by the formula of Eiscnbud -Levine [6]. This has recently been extended to more general real curve singularities by Montaldi and van Stratcn [I I]. In the case of complex curve singularities the original formula of Milnor [ IOJ for plant curve singularities p = 26 r + I has been gcncralized by Buchweitz ~Grcucl[3J to gcncral curve singularities. As is often the case a gcncral formula about invariants of singularities can bc rcplaccd by a much simpler formula in the wcightcd homogcncous cast. In this note wc give such a formula for the number of branches for real or complex curves in terms of the Jacobian algebra of the curve singularity. Recall that for/above, the Jacobian algebra J(j) = I’$ + , /J (1). whcrc E, + , dcnotcs the algchra of smooth germs on R”+‘, 0 and J(j’) dcnotcs the Jacobian idcal gcncratcd by the coordinate functionsj; of/. together with the n x n minors of the n x n + I matrix ((lj;/?.~,). The basic observation that suggests there may be a simpler formula whcnf is wcightcd homogeneous is that in this case (with/defining an isolated singularity, so J(j) is a finite dimensional vector space) J(j) is a O-dimensional Gorenstein algebra. This means there is a linear functional 4: J(j)R such that the bilinear multiplication pairing J(j) x J(/) + R given by (g.h.)H b(g.h) is nonsingular. For complete intersection curves this was shown in [4] to be equivalent to a duality result for surface singularities, which was subsequently proven by Wahl [ 133. Later, Wahl defined a Jacobian algebra J(X) for arbitrary weighted homogeneous complex curve singularities (X. 0) which are Gorenstein and proved quite generally that the J(X) is O-dimensional Gorenstein [I43 (also see [93). This Jacobian algebra has a middle weight s so that 2s is the maximal nonzcro weight in J(X). We Ict J(X),, resp. J(X),,, resp. J(X),, denote the subspaces of J(X) generated by elements of weight = s, resp. s. Since the multiplication pairing is additive on weights, it restricts to a nonsingular pairing on J(X), and dually pairs J(X),, with J(X),,. The pairing in the real case has the signature as an invariant; we shall denote it by sig(J(/)) (or sig(J(X))). By the above remark sig(J(X)) = sig(J(X),); and hence, in particular, also for J(f). The original Eisenbud-Levine result computed the degree of