Moduli of irreducible plane curve singularities with the semigroup \(<a,b>\). (English) Zbl 0617.14021

Algebraic geometry, Proc. Conf., Berlin 1985, Teubner-Texte Math. 92, 236-258 (1986).
[For the entire collection see Zbl 0607.00004.]
The authors construct and study the moduli space of irreducible plane curve singularities with the semigroup \(<a,b>\) (where a and b are positive relatively prime integers) and with minimal Tjurina number. More precisely, let \((X_ 0,0)\) be the analytic germ in \(({\mathbb{C}}^ 2,0)\) given by \(X^ a+Y^ b=0\). Let \(X\to H\) be a good representative of the versal \(\mu\)-constant deformation (where \(\mu\) is the Milnor number of \((X_ 0,0))\). It turns out that H can be chosen to be \({\mathbb{C}}^ n\) for some n, and X a hypersurface in \(H\times {\mathbb{C}}^ 2\). Then the moduli space is obtained as the quotient of H by \(G=\exp (V)\), where V is the kernel of the Kodaira-Spencer map. A good quotient space exists only if one fixes the Tjurina number. A special attention is payed to the quotient space corresponding to the minimal Tjurina number, in which case it is a geometric quotient in the sense of Mumford, and moreover, one shows that it is quasi-smooth and one computes its dimension explicitly.
Reviewer: L.Bădescu


14H10 Families, moduli of curves (algebraic)
14H20 Singularities of curves, local rings
14B05 Singularities in algebraic geometry


Zbl 0607.00004