Moduli of representations of one-point extensions

TL;DR

This paper investigates the geometric and algebraic properties of moduli spaces of representations of one-point extensions of quivers, unifying various classical moduli spaces and providing new criteria and formulas.

Contribution

It introduces a comprehensive study of moduli spaces for one-point extensions, including criteria for non-emptiness, geometric properties, semi-invariants, and cohomological formulas.

Findings

Numerical criteria for non-emptiness of moduli spaces

Construction of generating semi-invariants

Formula for Poincare polynomial in singular cohomology

Abstract

We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincare polynomial in singular cohomology of these moduli spaces.