The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb, while the curve graph, which collapses subgraphs corresponding to isotopy classes, was proven to be hyperbolic by Masur–Minsky. In this talk, we prove that subgraphs of the fine curve graph corresponding to curves that essentially intersect a common curve contain a quasi-isometrically embedded flat of every dimension and therefore are not hyperbolic. In particular, the subgraph of the fine curve graph induced by any single isotopy class—a graph whose properties are captured by neither the curve graph nor fine curve graph—is not hyperbolic. This is joint work with Ryan Dickmann.