Resumen de A note on Harris Morrison sweeping families

Valentina Beorchia, Francesco Zucconi

• Harris and Morrison (Invent. Math. 99:321–355, 1990, Theorem 2.5), constructed certain semistable fibrations ?:?→? in ? -gonal curves of genus ? , such that for every ? the corresponding modular curves give a sweeping family in the ? -gonal locus ??⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ . Their construction depends on the choice of a smooth curve ? . We show that if the genus ?(?) is sufficiently high with respect to ? , then the ratio ?2??(?) is 8 asymptotically with respect to ?(?) . Moreover, if the conjectured estimates given in Harris and Morrison (Invent. Math. 99:321–355, 1990, pp. 351–352) hold, we show that if ? is big enough, then ? is a surface of positive index.