Notes On Fundamental Groupoid Schemes

TL;DR

This paper investigates the fundamental groupoid schemes associated with Tannakian categories of vector bundles, computing specific cases like anisotropic conic, Klein bottle, and abelian varieties, and explores their interrelations.

Contribution

It provides explicit computations of fundamental groupoid schemes for certain geometric objects and analyzes their relationships through representation theory.

Findings

Computed fundamental groupoid schemes for anisotropic conic, Klein bottle, and abelian varieties.

Analyzed the relations among different fundamental groupoid schemes via their representations.

Enhanced understanding of the structure of fundamental groupoid schemes in algebraic geometry.

Abstract

In this article, we study the various fundamental groupoid schemes corresponding to Tannakian categories of certain types of vector bundles. We compute fundamental groupoid scheme of anisotropic conic, Klein bottle and abelian varieties. Additionally, we study the relation among various fundamental groupoid schemes by considering their representations.