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

Abstract

Let F⊂ P²× P^2v be the 3-dimensional flag. Let π₁ F→ P² and π₂ F→ P^2v 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× P^2v. In this paper we study the bigraded Hilbert function H⁰(I_A(a,b)), (a,b)∈N², for a general A∈M(u,v). We also give geometric properties of I_A(a,b) (spannedness and a uniqueness result for non-general A∈ M(u,v)).