The complex of curves is a powerful tool for studying mapping class groups, surfaces, and 3-manifold topology. Masur and Minsky showed in 1999 that the curve complex is a delta-hyperbolic space, with delta possibly depending on the topology of the underlying surface. In around 2005, Bowditch reproved this result and obtained quantitative control on the hyperbolicity constant as a function of genus and number of punctures; delta grows at most logarithmically with complexity. In a forthcoming paper, we show that delta can be taken to be independent of genus, i.e., there exists some k so that all curve complexes corresponding to closed orientable surfaces are k-hyperbolic. The main idea of the proof is encapsulated by the following question: how many times must two simple closed curves intersect in order to be distance at least k in the curve graph of the genus g surface?