A Viro theorem without convexity hypothesis for trigonal curves

Abstract

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision.It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants.Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.

Topological types of real regular Jacobian elliptic surfaces

“…Hence, to a real trigonal curve C corresponds a colored graph on P 1 (C) verifying the above properties. Conversely, drawing dessins d'enfants allows to construct real trigonal curves with a given real scheme (see, for example, [20], [2]). Here we will only use a local description of the graph Γ that will allow us to collapse independently the ovals.…”

“…A classical argument explained below shows that all triangulations we used are convex. Note that this is in fact unnecessary due to [2] which proves that the convexity hypothesis can be dropped in the combinatorial patchworking theorem when starting from a triangulation of [(0, 0), (3n, 0), (0, 3)]. Take a triangulation of T by segments starting from (0, 3) and ending to points of [(0, 0), (6k, 0)].…”

“…Though this statement can be easily deduced from [20], we give here a detailed proof for the reader's convenience (see Proposition 4.3). The emergence of dessins d'enfants in real algebraic geometry is recent [18,20] (see also, for example, [2,3]). It is S. Orevkov [20] who discovered that dessins d'enfants can be used to construct real trigonal curves with given real scheme.…”

“…Bertrand and E. Brugalle [2] showed how one can recover the dessin d'enfant of a trigonal curve constructed by the combinatorial patchworking.…”

A nonalgebraic patchwork

“…In [1] we proved that in the case of curves of bidegree (3, 0) in n , the patchworked pseudoholomorphic curve is always isotopic to a real algebraic one in the same homology class.…”