A note on Harris Morrison sweeping families

Harris and Morrison construct semistable families 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 of 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 geographical slope is 8 asymptotically with respect to g(X). Moreover, if some conjectured estimates given by Harris and Morrison hold, we show that if g is big enough, then F is a surface of positive index.