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The Euler number of the normalization of an algebraic threefold with ordinary singularities

Shoji Tsuboi (2004)

Banach Center Publications

By a classical formula due to Enriques, the Euler number χ(X) of the non-singular normalization X of an algebraic surface S with ordinary singularities in P³(ℂ) is given by χ(X) = n(n²-4n+6) - (3n-8)m + 3t - 2γ, where n is the degree of S, m the degree of the double curve (singular locus) $D_{S}$ of S, t is the cardinal number of the triple points of S, and γ the cardinal number of the cuspidal points of S. In this article we shall give a similar formula for an algebraic threefold with ordinary singularities...

The incidence class and the hierarchy of orbits

László Fehér, Zsolt Patakfalvi (2009)

Open Mathematics

R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar{η}$ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ $\bar{η}$ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity...

The lattices and Möbius functions of stable closed subrootsystems and Hyperplane complements for classical Weyl Groups.

Peter Fleischmann, Ingo Janiszczak (1991)

Manuscripta mathematica

The number of critical points of a product of powers of linear functions.

Peter Orlik, Hiroaki Teralo (1995)

Inventiones mathematicae

Topologie des fonctions régulières et cycles évanescents.

Thomas Brélivet (2003)

Revista Matemática Complutense

One has two notions of vanishing cycles: the Deligne's general notion and a concrete one used recently in the study of polynomial functions. We compare these two notions which gives us in particular a relative connectivity result. We finish with an example of vanishing cycle calculation which shows the difficulty of a good choice of compactification.