关于最大角度的Harris Morrison扫描族的一个注记
A note on Harris Morrison sweeping families of maximal gonality
摘要：
Harris和Morrison构造了亏格为g的k-模曲线的半稳定族f：F \to Y，使得对于每一个k，对应的模曲线在模空间的k-模轨迹中给出一个扫描族。它们的构造取决于平滑曲线X的选择。我们证明了如果亏格g（X）相对于g足够高，则K_F^2 / χ（O_F）相对于g（X）渐近地为8.我们还证明了，如果角度是最大的，并且Harris和Morrison的一些修正估计成立，则纤维化f：F \to Y的斜率关于g渐近为12，并且F是正指数曲面.
英文摘要：
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 / χ(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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