The cohomology ring of certain compactified Jacobians

We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q\mathrm{/}p}$ of the rational curve $C_{q\mathrm{/}p}$ with planar equation $x^q=y^p$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $S{p}_{q\mathrm{/}p}$ in the affine flag variety of $S{L}_p$. We show that the perverse filtration on the cohomology of $J_{q\mathrm{/}p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps ${\mathbb{P}}^1\to C_{q\mathrm{/}p}$. We also propose several conjectures about $J_{q\mathrm{/}p}$ and more general compactified Jacobians.