Let Bbbk be an algebraically closed field, Q a finite quiver, and denote by textup {rep}_{Q}^{mathbf {d}} the affine Bbbk -scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set {mathcal {O}}_{M} of points in Bbbk isomorphic as representations to M is an orbit under an action on textup {rep}^{mathbf {d}}_{Q}Bbbk of a product of general linear groups. The orbit {mathcal {O}}_{M} and its Zariski closure overline {mathcal {O}}_{M}, considered as reduced subschemes of textup {rep}_{Q}^{{mathbf {d}}}, are contained in an affine scheme {mathcal {C}}_{M} defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of overline {{mathcal {O}}}_{M} and {mathcal {C}}_{M} coincide, or equivalently, overline {{mathcal {O}}}_{M} is the reduced scheme associated to {mathcal {C}}_{M}. Moreover, overline {mathcal {O}}_{M}={mathcal {C}}_{M} provided Q is a Dynkin quiver of type {mathbb {A}}, and this equality is a conjecture for the remaining Dynkin quivers (of type mathbb {D} and {mathbb {E}}). Let Q be a Dynkin quiver of type mathbb {D} and M a finite dimensional representation of Q. We show that the equality T_{N}overline {mathcal {O}}_{M}=T_{N}{mathcal {C}}_{M} of Zariski tangent spaces holds for any closed point N of overline {mathcal {O}}_{M}. As a consequence, we describe the tangent spaces to overline {mathcal {O}}_{M} in representation theoretic terms.