## Moduli of plane curve singularities with a single characteristic exponent.(English)Zbl 0909.14015

This paper studies the moduli space for irreducible germs of analytic plane curves with a single characteristic exponent, $$m/n.$$ Here $$m,n$$ are relatively prime, positive integers with $$n<m.$$ The simplest such curve is equivalent to the $$(n,m)$$-quasihomogeneous curve: $$y^n-x^m=0.$$ If such a curve is non-quasihomogeneous then it has a parametric representation of the form: $\begin{cases} x= t^n\\ y= t^m+bt^{m+s}+\sum_{i>s}b_{m+i}t^{m+i},\;b\in\mathbb{C}^*. \end{cases}$ such that neither $$m+s$$ nor $$n+s$$ belong to the semi-group generated by $$m$$ and $$n.$$ The integer $$s$$ is an invariant of the germ. This paper studies the stratification of the moduli space of germs of irreducible plane curves with a single characteristic exponent defined by this invariant. The results in the paper include:
1. It is shown that there are only finitely many possible values for the invariant $$s.$$ A simple description is given for a universal family containing all germs with characteristic exponent, $$m/n$$ and invariant $$s.$$
2. If the germ is given by the equation, $$f(x,y)=0$$ then the Tjurina number is defined by $\tau=\dim_{\mathbb C}[{\mathbb C}\{x,y\}/(f,f_x,f_y)].$ An algorithm is given to compute the Tjurina number for Zariski open subsets from each of the universal families defined in 1.
3. Let $${\mathcal M}_{(n,m)}$$ denote the moduli space of germs with characteristic exponent, $$m/n.$$ It is shown that the subsets, $${\mathcal M}_{(n,m,s)}$$ with invariant equal to $$s$$ are connected and locally closed and define a stratification of $${\mathcal M}_{(n,m)}.$$ On each stratum the Tjurina number generically assumes its minimum value, $$\tau_{\min}(n,m,s).$$
4. Using deformation theory it is shown that $\dim{\mathcal M}_{(n,m,s)}=N_s+1-\mu+\tau_{\min}(n,m,s).$ Here $$N_s=\#\{(i,j)\in{\mathbb N}^2| ni+mj>mn+s, 0\leq i\leq m-2, 0\leq j\leq n-2\}$$ and $$\mu=(n-1)(m-1)$$ is a Milnor number.

### MSC:

 14H20 Singularities of curves, local rings 32S10 Invariants of analytic local rings

CoCoA
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### References:

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