A note on Harris Morrison sweeping families

• Autores: Valentina Beorchia, Francesco Zucconi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 2, 2015, págs. 191-202
• Idioma: inglés
• DOI: 10.1007/s13348-014-0128-5
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• Resumen

  • 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.