A Remark on Classical Pluecker’s formulae

Vik.S. Kulikov

doi:10.5802/afst.1517

Vik.S. Kulikov ¹

¹ Steklov Mathematical Institute

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 959-967.

Abstract
Abstract

For any reduced curve $C \subset ℙ^{2}$ , we introduce the notions of the number of its virtual cusps $c_{v}$ and the number of its virtual nodes $n_{v}$ . We prove that the numbers $c_{v}$ and $n_{v}$ are non-negative and if $C$ is a curve with only ordinary cusps and nodes as its singular points, then $c_{v}$ is the number of its ordinary cusps and $n_{v}$ is the number of its ordinary nodes. In addition, if $\hat{C}$ is the dual curve of an irreducible curve $C$ and ${\hat{n}}_{v}$ and ${\hat{c}}_{v}$ are the numbers of its virtual nodes and virtual cusps, then the integers $c_{v}$ , $n_{v}$ , ${\hat{c}}_{v}$ , ${\hat{n}}_{v}$ satisfy classical Plücker’s formulae.

Pour toute courbe réduite $C \subset ℙ^{2}$ , on introduit la notion de nombre des points de rebroussement (cusps) virtuels $c_{v}$ et celle de nombre des points doubles ordinaires (nodes) virtuels $n_{v}$ . Ces deux nombres sont positifs ou nuls et ils coïncident avec le nombre des points singuliers du type respectif lorsque ce sont les seules singularités de la courbe. De plus, si $\hat{C}$ est la courbe duale d’une courbe irréducible $C$ , et si ${\hat{n}}_{v}$ et ${\hat{c}}_{v}$ designent le nombre de singularités virtuelles de $\hat{C}$ du type respectif, alors les nombres entiers $c_{v}$ , $n_{v}$ , ${\hat{c}}_{v}$ , ${\hat{n}}_{v}$ vérifient les formules de Plücker classiques.

Published online: 2016-11-13

MR Zbl

DOI: 10.5802/afst.1517

Author's affiliations:

Vik.S. Kulikov ¹

¹ Steklov Mathematical Institute

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     title = {A {Remark} on {Classical} {Pluecker{\textquoteright}s} formulae},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {959--967},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
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Vik.S. Kulikov. A Remark on Classical Pluecker’s formulae. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 959-967. doi : 10.5802/afst.1517. https://afst.centre-mersenne.org/articles/10.5802/afst.1517/

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