A homological interpretation of the transverse quiver Grassmannians

TL;DR

This paper provides a homological characterization of the transverse quiver Grassmannian for affine quivers, showing it coincides with the smooth locus of certain irreducible components, thus linking geometric and homological aspects.

Contribution

It establishes a homological description of the transverse quiver Grassmannian as the set of points with vanishing Ext^1, connecting it to the smooth locus of minimal dimension components.

Findings

Transverse quiver Grassmannian equals the set of points with Ext^1 vanishing.

It coincides with the smooth locus of minimal dimension components.

Provides a homological interpretation linking geometry and representation theory.

Abstract

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^1(N,M/N)=0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.