  Harris and Morrison constructed semistable families f:F \to 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 K_F^2 / \chi(O_F) 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\to Y is 12 asymptotically with respect to g and that F is
a surface of positive index.
