Hwang and To on injectivity radius and gonality, and “Typical curves are not typical.”

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?