Link theory and oval arrangements of real algebraic curves
type: Item
instance of: scholarly article
title: Link theory and oval arrangements of real algebraic curves
zbMATH Open document ID: 0923.14032
DOI: 10.1016/S0040-9383(98)00021-4
published in: Topology
publication date: 1999-10-18T00:00:00Z
review text: This article is a significant contribution to the first part of Hilbert's 16th problem, topology of real plane algebraic curves. It contains a new method of obtaining restrictions to the disposition of branches of a real algebraic curve in the plane. The idea of the method is as follows. The union \(\Lambda_p\) of the complex points of real straight lines through a point \(p\in{\mathbb{R}}\mathbb{P}^2\) is a three-dimensional manifold which cuts \({\mathbb{C}}\mathbb{P}^2\) into two parts. Given the complexification \(A\subset{\mathbb{C}}\mathbb{P}^2\) of a real nonsingular algebraic curve, the intersection \(\Lambda_p \cap A\) is a union of intersecting circuits. Pushing \(\Lambda_p\) into one of the components of \({\mathbb{C}}\mathbb{P}^2\backslash\Lambda_p\), one turns \(\Lambda_p\cap A\) into a link \(L\) in \(S^3\) bounding an embedded surface \(N\subset B^4\) (which is, in fact, a piece of \(A\)). The link theory applies in this situation and gives necessary conditions for the existence of \(N\) such that \(\partial N=L\), in particular, a rather strong restriction to \(N\) in terms of the Seifert form of \(L\) is provided by the Murasugi-Tristram inequality. This stuff, completed with the quasipositivity of braids corresponding to real algebraic curves, results in a series of concrete statements on the arrangement of ovals of real plane curves. Among these results are an essential progress in the classification of affine M-curves of degree 6, projective reducible curves of degree 7, singular curves of degree 8 and nonsingular curves of degree 9, a new formula for complex orientations of a projective M-curve with a deep nest, a construction of a flexible M-curve of degree 8. The author provides a detailed description of an algorithm for computation of the Seifert form invariants of links and braids coming from real algebraic curves, and shows how it works in the above examples. The reviewer was informed by the author that the proofs of corollary 1.8 and lemma…
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 1307852
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: Seifert form of a link
zbMATH Keywords: Murasugi-Tristram inequality
zbMATH Keywords: quasipositive braids
zbMATH Keywords: topology of real plane algebraic curves
zbMATH Keywords: M-curves
zbMATH Keywords: degree
zbMATH Keywords: complex orientations
author: Stepan Orevkov
reviewed by: Eugenii I. Shustin
MaRDI profile type: Publication
full work available at URL: s0040-9383(98)00021-4
OpenAlex ID: W2119264916
description: scientific article; zbMATH DE number 1307852