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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

Software:

CoCoA
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References:

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[9] Oscar Zariski, Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 781 – 786.
[10] Oscar Zariski, Le problème des modules pour les branches planes, École Polytechnique, Paris, 1973 (French). Cours donné au Centre de Mathématiques de l’École Polytechnique, Paris, Octobre-Novembre 1973; Rédigé par François Kmety et Michel Merle; Avec un Appendice de Bernard Teissier et une réimpression de ”Characterization of plane algebroid curves whose module of differentials has maximum torsion” (Proc. Nat. Acad. Sci. U.S.A. 56 (1966), no. 3, 781 – 786), par Oscar Zariski.
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