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 $d \geq 1$ is an integer, $0 \leq \beta \leq 1$, 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.