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Salvetti complex, spectral sequences and cohomology of Artin groups

Filippo Callegaro (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

Springer fiber components in the two columns case for types A and D are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve C and the stability of the sheaf of logarithmic vector fields along C , the freeness of the divisor C and the Torelli properties of C (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

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