A note on Harris Morrison sweeping families of maximal gonality

TL;DR

This paper analyzes Harris and Morrison's construction of sweeping families of k-gonal curves, showing asymptotic ratios of invariants and implications for the slope and positivity of the resulting surfaces.

Contribution

It demonstrates that for high genus base curves, the ratio K_F^2 / χ(O_F) approaches 8, and under certain conjectures, the slope approaches 12, revealing new asymptotic properties.

Findings

K_F^2 / χ(O_F) asymptotically approaches 8 for high genus X.

The slope of the fibration approaches 12 under conjectured estimates.

The surface F has positive index in the asymptotic regime.

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.