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

摘要

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 ${h^0(mathcal{I}_X(t)) = 0}$ .