Unions of lines in ${\mathbb{R}}^n$

We prove a conjecture of D. Oberlin on the dimension of unions of lines in ${\mathbb{R}}^n$. If $d\ge 1$ is an integer, $0\le \beta \le 1$, and $L$ is a set of lines in ${\mathbb{R}}^n$ with Hausdorff dimension at least $2(d-1)+\beta$, then the union of the lines in $L$ has Hausdorff dimension at least $d+\beta$. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear to linear argument of Bourgain and Guth.