## 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

### MSC:

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

Zbl 0607.00004