Universal family of the subgroups of an algebraic group

TL;DR

This paper constructs a universal moduli space for connected subgroups of algebraic groups, enabling parametrization of flat families and analysis of non-connected subgroup families within algebraic geometry.

Contribution

It introduces a universal moduli space for connected algebraic subgroups and describes the structure of families of non-connected subgroups and their identity components.

Findings

Constructed a moduli space for connected subgroups of algebraic groups.

Established the existence of quotients of subgroup families by their identity components.

Described the structure of families of non-connected subgroups.

Abstract

We construct a moduli space for the connected subgroups of an algebraic group and the corresponding universal family. Morphisms from an algebraic variety to this moduli space correspond to flat families of connected algebraic subgroups parametrised by this variety. This moduli space is obtained by gluing together infinitely many irreducible projective varieties of bounded dimension along closed subvarieties. Regarding families of non-connected subgroups of an algebraic group, we show that, given sich a family, the corresponding family of identity components is an irreducible component of the former, and the quotient of a family of groups group by the family of their identity components exists.