Cech cover of the complement of the discriminant variety. Part II: Deformations of Gauss-skizze

TL;DR

This paper explores the topological and algebraic structures of the configuration space of points on the complex plane using Gauss-skizze graphs, revealing new deformation and operad insights related to Frobenius manifolds.

Contribution

It introduces a semi-algebraic stratification of the configuration space via Gauss-skizze, and links deformation theory to Hamilton-Jacobi equations, expanding the understanding of related operads.

Findings

Gauss-skizze form a cell decomposition of the configuration space.

Deformations governed by a Hamilton-Jacobi equation.

Introduces a Gauss-skizze operad related to the little 2-disc operad.

Abstract

The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton--Jacobi differential equation. This gives developments concerning Saito's Frobenius manifold. Finally, a Gauss-skizze operad is introduced. It is an enriched Fulton--MacPherson operad, topologically equivalent to the little 2-disc operad.