Le cerf-volant d'une constellation

TL;DR

This paper introduces a geometric simplicial complex called a kite to represent constellations of infinitely near points on complex surfaces, unifying various graph representations and linking them to the valuative tree.

Contribution

It constructs a geometric kite that contains both Enriques diagrams and dual graphs, providing a clear geometric interpretation of their relations and connections to the valuative tree.

Findings

The kite encodes the combinatorics of constellations.

It reveals geometric relations between different graph representations.

The structure links to continued fractions and lattice complexes.

Abstract

Consider a smooth point O of a complex analytic surface S. A constellation based at O is a set of infinitely near points of O, centers of a sequence of blow-ups above O. Finite constellations are usually encoded in two ways: either using an Enriques diagram, or using the dual graph of the divisor obtained by blowing-up the points of the constellation. Both are decorated trees which encode completely the combinatorics of the constellation. Algorithms of passage from one to the other are known, but they do not allow to get a geometrical picture of their relation. We associate to a constellation a geometrical simplicial complex of dimension two, called its kite, endowed with an affine structure, and we prove that it contains canonically both the Enriques diagram and the dual graph. Moreover, the decorations of the two trees may be read very easily on the affine geometry of the kite. This allows to understand geometrically the relations between the graphs, as well as their relation with the valuative tree of Favre and Jonsson, which may be interpreted as the dual graph of the constellation of all the points infinitely near O. In fact, the kites of finite constellations get glued into an infinite kite endowed with a 1-dimensional foliation, whose space of leaves is the valuative tree. The transition towards the computations with continued fractions is ensured by partial embeddings of the kites into a simplicial complex canonically associated to a base of a lattice, called its lotus. This last notion is briefly explored in any dimension.