Polar singularity of a flat curve germ

• Grant number: 17/15369-0
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: November 01, 2017
• End date: May 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Marcelo Jose Saia
• Grantee: Benoit Antoine Guerville
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM

Abstract

Let (f) be a germ of a flat irreducible curve on C. The description of the singularity of the polar of f is an open problem since the time of Max Noether. Results obtained by: Zariski, Casas Alvero, R. Peraire, Hefez and Hernandes, among others show that we can carry out a study of the polar using the classification of curves, topological properties of the Jacobian ideal and Newton's polynomial). The problem of the classification of the curves began with the work of Ebey and Zariski and was recently completed in the work of Hefez and Hernandez. Another question involving the plane algebraic curves is the description of its topology, which if considered as an abstract set is determined by its combinatorial data, the embeddings of these curves in the complex projective plane are still not well understood, Zariski was the first to prove that this type of homeomorphism is not determined by the combinatorial data of the curve. We intend to understand this gap between the embeding of a curve in the complex projective plane and its the combinatorial data. A special case of plane curves are the line arrangements, a curve whose all irreducible components are lines. The study of line arrangements admits many facets: combinatorial, algebraic, geometric or topological.

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BARTOLO, ENRIQUE ARTAL; GUERVILLE-BALLE, BENOIT; VIU-SOS, JUAN. Fundamental Groups of Real Arrangements and Torsion in the Lower Central Series Quotients. EXPERIMENTAL MATHEMATICS, v. 29, n. 1, p. 28-35, MAR 2 2020. (17/15369-0, 16/14580-7)

GUERVILLE-BALLE, BENOIT; VIU-SOS, JUAN. Configurations of points and topology of real line arrangements. MATHEMATISCHE ANNALEN, v. 374, n. 1-2, p. 1-35, JUN 2019. (17/15369-0, 16/14580-7)

GUERVILLE-BALLE, BENOIT. The loop-linking number of line arrangements. MATHEMATISCHE ZEITSCHRIFT, v. 301, n. 2, p. 30-pg., 2022-01-31. (17/15369-0)

GUERVILLE-BALLE, BENOIT. TOPOLOGY AND HOMOTOPY OF LATTICE ISOMORPHIC ARRANGEMENTS. Proceedings of the American Mathematical Society, v. 148, n. 5, p. 2193-2200, MAY 2020. (17/15369-0)