A geometric version of BGP reflection functors

TL;DR

This paper introduces a geometric approach to quiver Grassmannians and flags, establishing conditions for their smoothness and irreducibility, and develops a geometric version of BGP reflection functors to connect Hall algebras and Reineke's monoid.

Contribution

It presents a geometric version of BGP reflection functors and links Hall algebras at q=0 with Reineke's monoid in the Dynkin case.

Findings

Quiver Grassmannians are smooth and irreducible under certain conditions.

Counting polynomials have positive Euler characteristics and specific evaluations.

Geometric BGP reflection functors lead to algebraic isomorphisms.

Abstract

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to one of Reineke. Using this we can show that if there is a generic representation, then these varieties are smooth and irreducible. If we additionally have a counting polynomial we deduce that their Euler characteristic is positive and that the counting polynomial evaluated at zero yields one. After having done so, we introduce a geometric version of BGP reflection functors which allows us to deduce an even stronger result about the constant coefficient of the counting polynomial. We use this to obtain an isomorphism between the Hall algebra at q=0 and Reineke's generic extension monoid in the Dynkin case.