Picard groups and the K-theory of curves with cuspidal singularities

We calculate the algebraic $K$-theory of the coordinate ring of a planar cuspidal curve over a regular $\mathbb{F}_p$-algebra, thereby verifying a conjecture due to Hesselholt. In the course of the proof we compute the Picard group of the homotopy category of $p$-complete genuine $C_{p^n}$-spectra.