Irreducible components of module varieties: projective equations and rationality

TL;DR

This paper develops new methods to analyze the irreducible components of module varieties by linking affine and projective parametrizations, providing criteria for geometric properties and equational descriptions.

Contribution

It introduces a framework connecting $Rep(A,\bold d)$ and $GRASS(A,\bold d)$ to study irreducible components, including conditions for properties like unirationality, smoothness, and normality.

Findings

Criteria for unirationality, smoothness, and normality of components

Equational descriptions of irreducible components in the projective setting

Techniques for extracting qualitative geometric information

Abstract

We expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective variety, $GRASS(A,\bold d)$, parametrizing the same set of isomorphism classes of modules. In particular, we show the irreducible components to be accessible in a highly compressed format within the projective setting. Our results include necessary and sufficient conditions for unirationality, smoothness, and normality, followed by applications. Moreover, they provide equational access to the irreducible components of $GRASS(A,\bold d)$ and techniques for deriving qualitative information regarding both the affine and projective scenarios.