Virtual motives for synthetic geometries, A. Definition and properties of $K_0(\mathcal{Q}_\ell)$

TL;DR

This paper introduces Grothendieck rings for incidence geometries, focusing on generalized quadrangles, providing a new motivic framework to study synthetic geometry with novel properties and insights.

Contribution

It presents the first foundational work on Grothendieck rings for incidence geometries, establishing a motivic approach to synthetic geometry.

Findings

Development of Grothendieck rings for incidence geometries

Application to generalized quadrangles and related geometries

New properties and questions in synthetic geometry

Abstract

In this note, we introduce the first basics on Grothendieck rings for incidence geometries as a new motivic way and tool to study synthetic geometry. In this first instance, we concentrate on generalized quadrangles and related geometries. Many questions, new properties and insights arise along the way.