Hilbert–Kunz multiplicity and <i>F</i> ‐signature can disagree

Abstract We compute the ‐signature function of the ample cone of any nontrivial ruled surface over , where is an algebraically closed field of prime characteristic. As an application, we construct a Noetherian ‐finite strongly ‐regular ring of prime characteristic admitting two maximal ideals at which the Hilbert–Kunz multiplicity and ‐signature measure different singularities; that is, and . Our calculation of the ‐signature for the Hirzebruch surfaces also corrects an inaccuracy in a preprint by other authors.