Decorated trees

TL;DR

This paper introduces decorated trees as a combinatorial framework that generalizes various algebraic geometry diagrams, providing purely combinatorial proofs of formulas previously understood topologically.

Contribution

It defines decorated trees with specific decorations, extending formulas from algebraic geometry into a purely combinatorial context.

Findings

Recovered topological formulas combinatorially

Extended the applicability of algebraic geometry diagrams

Unified different types of trees under a common combinatorial framework

Abstract

We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and the decorations are required to satisfy certain conditions. The class of decorated trees includes different types of trees used in algebraic geometry, such as the Eisenbud and Neumann diagrams for links of singularities and the Neumann diagrams for links at infinity of algebraic plane curves. By purely combinatorial means, we recover some formulas that were previously understood to be "topological". In this way, we extend the generality of those formulas and show that they are in fact "combinatorial".