Complexity of Geometric k-Planarity for Fixed k Marcus Schaefer Vol. 25, no. 1, pp. 29-41, 2021. Regular paper. Abstract The rectilinear local crossing number, $\mathop{\overline{\rm lcr}}(G)$, of a graph $G$ is the smallest $k$ so that $G$ has a straight-line drawing with at most $k$ crossings along each edge. We show that deciding whether $\mathop{\overline{\rm lcr}}(G) \leq k$ for a fixed $k$ is complete for the existential theory of the reals, $\exists \mathbb{R}$.  This work is licensed under the terms of the CC-BY license. Submitted: August 2020. Reviewed: January 2021. Revised: January 2021. Accepted: January 2021. Final: January 2021. Published: January 2021. Communicated by Giuseppe Liotta article (PDF) BibTeX