A note on Harris Morrison sweeping families

Harris and Morrison (Invent. Math. 99:321–355, 1990, Theorem 2.5), constructed certain semistable fibrations f:F→Y in k-gonal curves of genus g, such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus (formula presented). Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio (formula presented) is 8 asymptotically with respect to g(X). Moreover, if the conjectured estimates given in Harris and Morrison (Invent. Math. 99:321–355, 1990, pp. 351–352) hold, we show that if g is big enough, then F is a surface of positive index.