zbMATH — the first resource for mathematics

Normal forms and moduli spaces of curve singularities with semigroup \(<2p,2q,2pq+d>\). (English) Zbl 0725.14021
The authors classify, over the complex numbers and modulo analytic transformations, maps and plane curve germs given by equations of the form \(f=(x^ p+y^ q)^ 2+\sum_{iq+jp>2pq}a_{ij}x^ iy^ j,\gcd (p,q)=1\) with prescribed Milnor number. This in particular includes the classification of all irreducible germs of plane curves with characteristic exponents \((p/q,(2d+1)/2q)\) for given p, q and d. The authors find the corresponding moduli spaces to be Zariski open sets of complex affine spaces modulo a suitable action of a group of roots of unity. Universal families are found and the Tjurina number \(\tau\) turns out to be constant in these cases.

14H20 Singularities of curves, local rings
14H10 Families, moduli of curves (algebraic)
Full Text: Numdam EuDML
[1] Laudal, O.A. , Martin, B. , and Pfister, G. , Moduli of irreducible plane curve singularities with the semi-group , Proc. Conf. Algebraic Geometry Berlin. Teubner-Texte 92 (1986). · Zbl 0617.14021
[2] Laudal, O.A. , and Pfister, G. , Local moduli and singularities , Springer, Lecture Notes 1310 (1988). · Zbl 0657.14005
[3] Mora, F. , A constructive characterization of standard bases , Boll. U.M.I. sez. D. 2 (1983) 41-50. · Zbl 0619.13010
[4] Zariski, O. , Le probleme des modules pour les branches planes , Ed. Hermann, Paris 1986. · Zbl 0592.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.