The bigraded Hilbert function of unions of lines in the 3-dimensional flag variety

Let F⊂ P2× P2v be the 3-dimensional flag. Let π1 F→ P2 and π2 F→ P2v be the projections. For all u,v ∈N\{(0,0)} let M(u,v) denote the set of all curves π1-1(F) ∪ π2-1(E) such that π1-1(F) ∩ π2-1(E)=∅, #F=v and #E=u. Any A∈ M(u,v) has u+v connected components, all of them smooth and rational and embedded as lines by the Segre embedding of F⊂ P2× P2v. In this paper we study the bigraded Hilbert function H0(IA(a,b)), (a,b)∈N2, for a general A∈M(u,v). We also give geometric properties of IA(a,b) (spannedness and a uniqueness result for non-general A∈ M(u,v)).