Curvettes and clusters of infinitely near points

TL;DR

This paper revises the theory of clusters of infinitely near points over arbitrary fields, detailing intersection matrices, introducing curvettes, and relating them to Hamburger-Noether tableaux for curves.

Contribution

It extends the theory of clusters and curvettes to arbitrary fields, providing new insights into their intersection properties and relations to classical tableaux.

Findings

Describes the intersection matrix of clusters over arbitrary fields

Introduces the notion of curvette in this context

Relates curvettes to Hamburger-Noether tableaux

Abstract

The aim of this paper is to revise the theory of clusters of infinitely near points for arbitrary fields. We describe in particular the intersection matrix of such a cluster, we introduce the notion of curvette over an arbitrary field and we relate it to the Hamburger-Noether tableaux associated with curves.