A note on Harris Morrison sweeping families of maximal gonality

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Harris and Morrison constructed semistable families f:F → Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. 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 KF² / χ(OF) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F→ Y is 12 asymptotically with respect to g and that F is a surface of positive index.