A note on Harris Morrison sweeping families of maximal gonality

Valentina Beorchia; Francesco Zucconi

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arXiv · 1302.0620

A note on Harris Morrison sweeping families of maximal gonality

Valentina Beorchia·Francesco Zucconi

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Abstract

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.

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Valentina Beorchia, Francesco Zucconi. 2013-02-04. A note on Harris Morrison sweeping families of maximal gonality. https://arxiv.org/abs/1302.0620

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