Let $G$ be an intersection graph of $n$ geometric objects in the plane. We show that a maximum matching in $G$ can be found in $O(ρ^{3ω/2}n^{ω/2})$ time with high probability, where $ρ$ is the density of the geometric objects and $ω>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^ω)$ time. The same result holds for any subgraph of $G$, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in $O(n^{ω/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, Ψ]$ can be found in $O(Ψ^6\log^{11} n + Ψ^{12 ω} n^{ω/2})$ time with high probability.