Postulation of Disjoint Unions of Lines and Double Lines in $${\mathbb{P}^3}$$

Abstract

A double line \({C \subset \mathbb{P}^3}\) is a connected divisor of type (2, 0) on a smooth quadric surface. Fix \({(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}\). Let \({X \subset \mathbb{P}^3}\) be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each \({t \in \mathbb{Z}}\) either \({h^1(\mathcal{I}_X(t)) = 0}\) or \({h^0(\mathcal{I}_X(t)) = 0}\).