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.