\(G\)-signature, \(G\)-degree, and symmetries of the branches of curve singularities (Q1184324)

Let \((X,0)\) be a real or complex curve singularity on which a finite group \(G\) acts. Then \(G\) acts on the set \(B\) of real or complex branches of \(X\). Picking any field \(\mathbf k\) there is a corresponding permutation representation \(V_ p\) of dimension \(r=\# B\). The author considers the problem of determining this representation. In the real case, the results relate the corresponding (modular) character \(\chi_ p\), or more precisely, a virtual character \(\chi_{vp}\) associated with the representation of \(G\) on the oriented half-branches of \(X\), to an algebraically computable \(G\)-signature. Building on result by \textit{K. Aoki, T. Fukuda} and \textit{T. Nishimura} [On the number of branches of the zero locus of a map germ \((\mathbb{R}^ n,0)\to (\mathbb{R}^{n-1},0)\): Topology and Computer Sciences, S. Suzuki ed. Kinokuniya Co. Ltd. Tokyo 347-367 (1987)] and generalizations by \textit{J. Montaldi} and \textit{D. van Straten} [Topology 29, No. 4, 501-510 (1990)] and the author [Topology 30, No. 2, 223-229 (1991; Zbl 0723.32014)], he proves that in \(\text{char }{\mathbf k}=2\), \[ \chi_{vp}=\text{sig}_ G(\psi_ \alpha^ +)+\text{sig}_ G(\psi^ -_ \alpha), \] where \(\alpha\) is a finite form, in the sense of Montaldi and van Straten, i.e. a real meromorphic 1-form different from zero on every branch of the complexified curve \(X_ C\), and where \(\psi_ \alpha^ +\) and \(\psi^ -_ \alpha\) are the nonsingular real symmetric bilinear forms defined on the real part of the ramification modules, \(R^ +_ \alpha=\omega_{X_ C}/O_{X_ C}\alpha\cap\omega_{X_ C}\), \(R_ \alpha^ -=O_{X_ C}\alpha/O_{X_ C}\alpha\cap\omega_{X_ C}\), by the restriction of the residue pairing \(\langle\omega_ 1,\omega_ 2\rangle=\text{Res}(h\omega_ 2)\), where \(\omega_ 1=h\alpha\). Using this result he can deduce generalizations of the result of Aoki, Fukuda and Nishimura, referred to above, and some more precise results for weighted homogeneous and semi-weighted homogeneous curve singularities, and for odd-ordered groups. These considerations also lead to the definition of a \(G\)-degree and a generalization of the result of \textit{D. Eisenbud} and \textit{H. I. Levine} on the degree of finite map germs [Ann. Math., II. Ser. 106, 19-44 (1977; Zbl 0398.57020)].