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

TL;DR

This paper investigates the Hilbert function of unions of lines and double lines in projective 3-space, establishing conditions under which these unions have maximal rank and providing specific examples where they do not.

Contribution

It determines when unions of lines and double lines in ^3 have maximal Hilbert function rank, extending understanding of their algebraic properties.

Findings

Unions with certain configurations have maximal Hilbert function rank.

Explicit conditions for maximal rank are provided for various numbers of lines and double lines.

Examples where the union does not have maximal rank are also given.

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.