The Hilbert function of general unions of lines and double lines in the projective space

Abstract

We study the Hilbert function of a general union $X\subset \mathbb{P}^3$ of $x$ double lines and $y$ lines. In many cases (e.g. always for $x=2$ and $y\ge 3$ or for $x=3$ and $y\ge 2$ or for $x\ge 4$ and $y\ge \lceil(\binom{3x+4}{3} -27x-12)/(3x+2)\rceil +3-x$) we prove that $X$ has maximal rank. We give a few examples of $x$ and $y$ for which $X$ has not maximal rank.