A homological interpretation of the transverse quiver grassmannians

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 Ext1(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.