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Given a plane curve singularity C, one can define an algebraic variety called the compactified Jacobian of C. It is closely related to a singular fiber of the Hitchin fibration and to certain orbital integrals in number theory. We introduce a class of "generic" curves, and describe the homology of the corresponding compactified Jacobians in terms of combinatorics of non-coprime rational q,t-Catalan numbers. All notions will be introduced in the talk, this is a joint work with Mikhail Mazin and Alexei Oblomkov.