Mass formula for non-ordinary curves in one dimensional families
This paper is about one dimensional families of cyclic covers of the projective line in positive characteristic. For each such family, we study the mass formula for the number of non-ordinary curves in the family. We prove two equations for the mass formula: the first relies on the $a$-numbers of non-ordinary curves in the family; and the second relies on tautological intersection theory. Our results generalize the Eichler--Deuring mass formula for supersingular elliptic curves; they also generalize some theorems of Ibukiyama, Katsura, and Oort about supersingular curves of genus $2$ that have an automorphism of order $3$ or order $4$. We determine the mass formula in many new cases, including linearized families of hyperelliptic curves of every genus and all families of cyclic covers of the projective line branched at four points. For certain families of curves of genus $3$ to $7$, we determine the exact rate of growth of the number of non-ordinary curves in the family as a linear function of the characteristic $p$. keywords: curve, hyperelliptic curve, cyclic cover, Jacobian, mass formula, cycle class, tautological ring, Hodge bundle, intersection theory, Frobenius, non-ordinary, $p$-rank, $a$-number.