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A bound for the Milnor number of plane curve singularities

Arkadiusz Płoski (2014)

Open Mathematics

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

Let ( 𝒮 , 0 ) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E i , i I . The Nash map associates to each irreducible component C k of the space of arcs through 0 on 𝒮 the unique component of E cut by the strict transform of the generic arc in C k . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if E · E i < 0 for any  i I .

A note on Bézout's theorem

Sławomir Rams, Piotr Tworzewski, Tadeusz Winiarski (2005)

Annales Polonici Mathematici

We present a version of Bézout's theorem basing on the intersection theory in complex analytic geometry. Some applications for products of surfaces and curves are also given.

A Note on the Rational Cuspidal Curves

Piotr Nayar, Barbara Pilat (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.

A propos du problème des arcs de Nash

Camille Plénat (2005)

Annales de l’institut Fourier

Soit = N i la décomposition canonique de l’espace des arcs passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que N i n’est pas inclus dans N j . On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.

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