A note on Harris Morrison sweeping families

Abstract

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.