Planar webs of maximum rank and analytic projective surfaces

Abstract

An analytic $d$-web $\mathcal{W}(d)$ in a neighborhood of $0\in {\mathbb C}^2$ with $d\ge 4$ and maximum rank $\pi _d:={1\over 2}(d-1)(d-2)$ gives rise to an analytic map ${\mathfrak u}:({\mathbb C}^2,0)\longrightarrow \check{\mathbb P}^{\,\pi _d-1}$ through an abelian basis. Classic and new properties of the resulting projective surface, smooth but transcendental in general, are presented. Modeled on the previous $\mathfrak u$, special maps $f:({\mathbb C}^2,0)\longrightarrow \check{\mathbb P}^{\,\pi _d-1}$ induce Bompiani webs $\mathcal{W}_f(d)$ through their $(d-4)$-principal curves that we define. It is proved in particular that those Bompiani webs $\mathcal{W}_f(d)$ of maximum rank are characterized only by a vanishing curvature.