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?

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This is cool. I’m kinda surprised by their Theorem 2. You get some lower bound on the largest injectivity radius among k-gonal curves by looking at hyperbolic orbifolds whose underlying space is a sphere and with cone angles bounded below. I guess the cone points are closer together than my intuition suggests. Neat-o.

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