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An upper bound of the singularity order for the generic projection. (English) Zbl 0673.14018
Let X be an algebroid curve of multiplicity e in $$(k^ N,0)$$, $$\tilde X$$ be Zariski’s absolute saturation of X and $$X'$$ a generic plane projection of X. The author proves that $$\delta (X')$$ is not bigger than $$(e-1)\delta(\tilde X)-(e-1)(e-2)/2,$$ where $$\delta$$ denotes as usual the codimension of the local ring of the singularity in its normalisation. Moreover the author proves that equality holds iff X is isomorphic to the “monomial curve” $$t\to (t^ e,t^{e+1},...,t^{2e-1})$$.
Reviewer: A.Buium

##### MSC:
 14H20 Singularities of curves, local rings 14E05 Rational and birational maps 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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##### References:
 [1] Campillo, A., Algebroid curves in positive characteristic, () · Zbl 0451.14010 [2] Serre, J.-P., Groupes algébriques et corps de classes, () · Zbl 0097.35604 [3] Zariski, O.; Zariski, O.; Zariski, O., General theory of saturations and saturated local rings, Amer. J. math., Amer. J. math., Amer. J. math., 97, 3, 415-502, (1975) · Zbl 0306.13009 [4] Zariski, O., Studies in equisingularity, Amer. J. math., 90, 3, 961-1023, (1968) · Zbl 0189.21405 [5] Zariski, O., Le problème des modules pour LES branches planes, (1973), Cours donné au Centre de Math. de L’Ecole Polytechnique Paris
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