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Moduli of germs of Legendrian curves. (English) Zbl 1187.14004

Legendrian curves (in \(3\)–space) are studied. The equisingularity type of a germ of Legendrian curve is (by definition) the topological type of its generic plane projection. If \(C\) is the germ at the origin of a singular irreducible plane curve parametrized by \(x=t^n, y=\sum_{i=m}^\infty a_it^i\) then the conormal \(\Lambda\) of the curve is parametrized by \( x=t^n, y=\sum_{i=m}^\infty a_i t^i\), \( p=\frac{dy}{dx}\cdot\sum_{i=m}^\infty\frac{i}{n}a_it^i\). This is an example for a Legendrian curve.
The generic component of the moduli space of the germs of Legendrian curves with generic plane projection topological equivalent to a curve \(y^n=x^m\) is constructed.

MSC:

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
14D22 Fine and coarse moduli spaces
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References:

[1] Arnold (V.I.).— First steps in local contact algebra, Can. J. Math. 51, No.6, p. 1123-1134 (1999). · Zbl 1031.53110
[2] Delorme (C.).— Sur les modules des singularités des courbes planes, Bull. Soc. Math. France 106, p. 417-446 (1978). · Zbl 0395.14010
[3] Greuel (G.-M.) and Pfister (G.).— Moduli for singularities. J.-P. Brasselet (ed.), Singularities, Lond. Math. Soc. Lect. Note Ser. 201, p. 119-146 (1994). · Zbl 0823.14001
[4] Kashiwara (M.) and Kawai (T.).— On holonomic systems of microdifferential equations. III: Systems with regular singularities. Publ. Res. Inst. Math. Sci. 17, p. 813-979 (1981). · Zbl 0505.58033
[5] Kashiwara (M.).— Systems of microdifferential equations. Progress in Mathematics, 34. Birkhauser. · Zbl 0521.58057
[6] Neto (O.).— Equisingularity and Legendrian curves, Bull. London Math. Soc. 33, p. 527-534 (2001). · Zbl 1032.58028
[7] Peraire (R.).— Moduli of plane curve singularities with a single characteristic exponent, Proc. Am. Math. Soc. 126, No.1, p. 25-34 (1998). · Zbl 0909.14015
[8] Zariski (O.).— Le problème des modules pour les branches planes. Hermann (1970). · Zbl 0592.14010
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