Projectivity of the moduli of equidimensional branchvarieties

TL;DR

This paper proves the projectivity of moduli spaces of equidimensional branchvarieties and constructs related projective moduli spaces with stability conditions, advancing the understanding of their geometric properties.

Contribution

It resolves an open problem on the projectivity of moduli of equidimensional branchvarieties and constructs new projective moduli spaces with stability conditions.

Findings

Proves the projectivity of the moduli of equidimensional branchvarieties.

Constructs projective moduli spaces of reduced equidimensional varieties with ample linear series.

Establishes a framework for moduli spaces with semistability conditions.

Abstract

We resolve an open problem posed by Alexeev-Knutson on the projectivity of the moduli of branchvarieties in the equidimensional case. As an application, we construct projective moduli spaces of reduced equidimensional varieties equipped with ample linear series and subject to a semistability condition.