Tissus linéaires et théorèmes d'algébrisation de type Abdel-inverse et Reiss-inverse

Abstract

A d-web in ($\mathbb{C}^n$,0) is given by d complex analytic foliations of codimension one in ($\mathbb{C}^n$,0) which are in general position. A d-web $\mathcal{L}(d)$ in ($\mathbb{C}^n$,0) is linear if all the leaves are (pieces of) hyperplanes in $\mathbb{C}^n$ and $\mathcal{L}(d)$ is algebraic if it is associated, by duality, to a nondegenerate algebraic curve Г in $\mathbb{P}^n$ of degree d. We characterize linear webs in ($\mathbb{C}^n$,0). We give explicit conditions under which a linear d-web in ($\mathbb{C}^n$,0) is algebraic and we obtain equations for $\Gamma \subset \mathbb{P}^n$ in this case. Some related problems are discussed and some questions are posed.