We consider minimizers of the energy Eε(u) =:∫Ω[1/2|(Δ)u|2+1/4ε2(|u|2-1)2]dx+1/2εs∫(δ)ΩW(u,g)ds,u∶Ω→C,0＜s＜1,in a two-dimensional domain Ω,with weak anchoring potential W(u,g) =:1/2(|u|2-1)2 + (-cosα)2,0 ＜ α ＜ π/2.This functional was previously derived as a thin-film limit of the Landau-de Gennes energy,assuming weak anchoring on the boundary favoring a nematic director lying along a cone of fixed aperture,centered at the normal vector to the boundary.In the regime where s[α2 + (π-α)2]＜ π2/2,any limiting map u* : α → S1 has only boundary vortices,where its phase jumps by either 2α (light boojums) or 2(π-α) (heavy boojums).Our main result is the fine-scale description of the light boojums.