Rokhlin's inequalities for real algebraic curves and an extremal property
type: Item
instance of: scholarly article
title: Rokhlin's inequalities for real algebraic curves and an extremal property
zbMATH Open document ID: 0952.14041
author: Stephen P. Paris
published in: Manuscripta Mathematica
publication date: 2001-01-15T00:00:00Z
review text: Let \(A\) be the real projective algebraic curve associated to a real homogeneous nonsingular polynomial of even degree \(2k\) whose complexification, which is denoted \(\mathbb{C} A\), is a submanifold of the complex projective plane \(\mathbb{C} \mathbb{P} (2)\) homeomorphic to a sphere with \(g= 2k^2- 3k+1\) handles. It is assumed that \(A\) is a dividing curve, i.e. \(\mathbb{C} A- A\) is disconnected. Let \(l\) be the number of ovals of \(A\) and let \(r= g+1-l\). Denote by \(p\) the number of even ovals of \(A\), and by \(n\) the number of odd ovals. A component of \(\mathbb{R} \mathbb{P} (2)- A\) is called even if each of the ovals bounding it from inside encloses an odd number of other ovals, and is called odd otherwise. The number of nonempty even ovals bounding an even component \(\mathbb{R} \mathbb{P} (2)- A\) from outside is denoted by \(\pi\), and the number of nonempty odd ovals bounding an even component of \(\mathbb{R} \mathbb{P}(2)- A\) from outside is denoted by \(\nu\). A famous result of \textit{V. A. Rokhlin} [Funct. Anal. Appl. 14, 29-33 (1980; Zbl 0478.14010)] states that if \(k\) is even, then \(n- 2\nu\geq \frac{3k-6-r} {2},\) while if \(k\) is odd, then \(p- 2\pi\geq \frac{3k-5-r} {2}.\) The extremal property referred in the title concerns those curves achieving equality in the first of the inequalities above. Namely, suppose \(A\) is oriented by complex conjugation, and let \(L\) be an oriented one-sided simple closed curve smoothly embedded in \(\mathbb{R} \mathbb{P} (2)- A\). Let \(\lambda^+(L)\) be the number of ovals of \(A\) which are oriented in the same direction as \(L\), \(\lambda^-(L)= 1- \lambda^+(L)\), and define \[ \omega= \omega(L)= [\lambda^+(L)- \lambda^-(L)]/ 2\pmod 2. \] An island of \(A\) is an outermost oval of \(A\) together with all ovals of \(A\) contained in it, and it is said to be even if it consists of an even number of ovals. The paper under review is devoted to give a new proof of the following result, already …
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 57R75
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 1310837
zbMATH Keywords: Rokhlin's inequalities
zbMATH Keywords: ovals
zbMATH Keywords: complex projective plane
zbMATH Keywords: dividing curve
zbMATH Keywords: number of ovals
zbMATH Keywords: link cobordism
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
full work available at URL: s002290050152
OpenAlex ID: W1971237058
DOI: 10.1007/S002290050152
description: scientific article; zbMATH DE number 1310837