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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.

##### MSC:
 14H20 Singularities of curves, local rings 14H10 Families, moduli of curves (algebraic)
##### Keywords:
constant Tjurina number; Milnor number; moduli spaces
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##### References:
 [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
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