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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.

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    $\begingroup$ I believe there is a conjecture that this is possible for gonality d iff d<g/3. I'll email the mathematician who told me this and ask for a reference. $\endgroup$
    – David Zureick-Brown
    Commented Feb 6, 2011 at 19:18
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    $\begingroup$ Slightly different question in very much the same spirit which I have asked many people and never got an answer: Is there ONE Brill-Noether generic curve defined over $\mathbb{Q}$ for every (or for infinitely many) genus $g$? $\endgroup$
    – Felipe Voloch
    Commented Feb 6, 2011 at 20:37
  • $\begingroup$ The only high-genus curves over Q I feel truly acquainted with are the modular curves, or Shimura curves more generally. Do they have unexpected linear systems? (Or rather: do we know they do? The gonality is certainly not known to be generic, though by work of Zograf and Abramovich it at least grows linearly in genus.) $\endgroup$
    – JSE
    Commented Feb 7, 2011 at 3:14
  • $\begingroup$ I can't prove offhand that modular curves are not Brill-Noether generic, but here are things one can try: There are maps to elliptic curves (which in turn are double covers of the projective line) which might give low degree maps. The j-map has degree linear with the genus (1/12 or 12?). Another candidate is the line bundle $\omega$, whose global sections are the modular forms of weight one and there might be just enough of them. $\endgroup$
    – Felipe Voloch
    Commented Feb 7, 2011 at 4:52

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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.

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