Quotient Spaces Modulo Algebraic Groups

TL;DR

This paper proves the existence of geometric quotients as algebraic spaces for proper actions of reductive group schemes, leading to the construction of moduli spaces of canonically polarized varieties over Spec Z.

Contribution

It establishes the existence of algebraic space quotients under proper reductive group actions, enabling new moduli space constructions over the integers.

Findings

Geometric quotients exist as algebraic spaces for proper reductive group actions.

Moduli spaces of canonically polarized varieties over Spec Z are constructed.

Provides foundational results for algebraic geometry and moduli theory.

Abstract

The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec Z.