We know that M\_g is general type for g large enough. In particular, the generic genus-g curve is not contained in a (non-isotrivial) rational family parametrized by P^1. In fact, the high-genus curves I know how to build over C(t) all have low gonality; it's easy to make a hyperelliptic curve y^2 = f(t,x), and with a little more work you can make curves over k(t) which are 3-gonal, 4-gonal, or 5-gonal. But do we know whether there's a curve over C(t) whose gonality is close to the generic value, which is on order g/2?


A colleague suggested that the linear system of the generator of Pic on a K3 with Picard number 1 would be a good place to look for these. I don't immediately see the proof that these guys have big gonality, but it certainly seems reasonable. 


So more generally, one might ask -- are there genus-g curves over C(t) which are "Brill-Noether generic," i.e. which are not distinguished from the generic genus-g curve by the presence of any g^r\_d?


Note that I am not asking "what do we expect the union of all rational curves on M\_g to look like, assuming we believe Lang's conjecture?" -- that question is too intimidating. Rather, I'm asking whether there is some Brill-Noether locus on M\_g which MIGHT contain all the rational curves. Having phrased it this way, I imagine this must be a question with some literature attached, but I wasn't able to find it.



Can a rational family of genus-g curves have generic gonality?  Can it be Brill-Noether general?
One can construct pencils of k-gonal curves of genus g by taking a K3 surface S with 
Pic(S) generated by two classes: an ample class C with self-intersection 2g-2, and an elliptic curve E, so self-intersection 0, such that C.E=k. Every curve in the linear system
|C| has gonality k, and the pencil of minimal degree is computed by the restriction of E to C (use for this the work of Green-Lazarsfeld). Here k varies from 2 to (g+2)/2, so you cut through all the gonality strata in this way.


These rational curves will not be covering the k-gonal locus M^1\_{g, k} inside M\_g. As to which of these loci are uniruled, what is known is that when g is large, the k-gonal locus becomes of general type when k exceeds g/3. It is not clear how optimal this bound is.


The follow-up question of Felipe, as to whether anyone has written down a Brill-Noether general curve over Q (and unbounded genus), is much more difficult, and as far as I am aware nobody knows.


