Classification of equivariantly normal curves via Altmann-Hausen-Süss theory

Let the ground field be perfect of positive characteristic. Using Altmann-Hausen-Süss theory, we obtain a combinatorial classification of equivariantly normal curves with prescribed quotient in both the affine and projective settings. As a consequence, we derive an explicit upper bound on the number of isomorphism classes of equivariantly normal curves over a fixed base curve with prescribed branch locus. Furthermore, assuming that the ground field is algebraically closed, we determine, for a fixed cardinality of the branch locus, precisely when the set of isomorphism classes of equivariantly normal projective curves with prescribed quotient is finite and when it is infinite.