The Max Noether Fundamental Theorem is Combinatorial

In the present paper we give a reformulation of the Noether Fundamental Theorem for the special case where the three curves involved have the same degree. In this reformulation, the local Noether's Conditions are weakened. To do so we introduce the concept of Abstract Curve Combinatorics (ACC) which will be, in the context of plane curves, the analogue of matroids for hyperplane arrangements

Integration of the kenzo system within sagemath for new algebraic topology computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before

An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the fundamental groups of their complements are not isomorphic (despite the fact that they have isomorphic profinite completions)

Around the tangent cone theorem

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.Comment: 39 pages; to appear in the proceedings of the Configurations Spaces Conference (Cortona 2014), Springer INdAM serie