3-dimensional sundials

Robin Hartshorne and Alexander Hirschowitz proved that a generic collection of lines on ${\mathbb{P}}^n$, $n\ge 3$, has bipolynomial Hilbert Function. We extended this result to a specialization of the collection of generic lines, by considering a union of lines and $3$-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).