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 a preprint by other authors.
Lee et al. (Wed,) studied this question.