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
• Texto completo no disponible (Saber más ...)
• Resumen

  • Harris and Morrison (Invent. Math. 99:321–355, 1990, Theorem 2.5), constructed certain semistable fibrations f:F\rightarrow Y in k-gonal curves of genus g, such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus \overline{\mathcal {M}^{k}_{g}}. 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 \frac{K^{2}_{F}}{\chi (\mathcal {O}_{F})} is 8 asymptotically with respect to g(X). Moreover, if the conjectured estimates given in Harris and Morrison (Invent. Math. 99:321–355, 1990, pp. 351–352) hold, we show that if g is big enough, then F is a surface of positive index.

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