Interesting new paper in the American Journal of Mathematics, not on arXiv unfortunately. An old theorem of Li and Yau shows how to lower-bound the gonality of a Riemann surface in terms of the spectral gap on its Laplacian; this (together with new theorems by many people on superstrong approximation for thin groups) is what Chris Hall, Emmanuel Kowalski, and I used to give lower bounds on gonalities in various families of covers of a fixed base.
The new paper gives a lower bound for the gonality of a compact Riemann surface in terms of the injectivity radius, which is half the length of the shortest closed geodesic loop. You could think of it like this — they show that the low-gonality loci in M_g stay very close to the boundary.
“The middle” of M_g is a mysterious place. A “typical” curve of genus g has a big spectral gap, gonality on order g/2, a big injectivity radius… but most curves you can write down are just the opposite.
Typical curves are not typical.
When g is large, M_g is general type, and so the generic curve doesn’t move in a rational family. Are all the rational families near the boundary? Gaby Farkas explained to me on Math Overflow how to construct a rationally parametrized family of genus-g curves whose gonality is generic, as a pencil of curves on a K3 surface. I wonder how “typical” these curves are? Do some have large injectivity radius? Or a large spectral gap?