Θ-functions on Hurwitz space loci

The data attached to Hurwitz spaces – Nielsen classes – generalize the notion of genus of a compact Riemann surface. Therefore, there is a natural map of any Hurwitz space into the moduli space of curves of the corresponding genus. 
  1. Expressing the analytic continuation of appropriate Θ functions along L.
  2. Evaluating expressions in these Θs to interpret aspects of curves along L.
  3. Interpreting global properties of L from the expressions in #2.
This part of the Theta-domain site is dedicated to papers that touch on aspects of this rubric. The subject is very classical. So, it is no surprise there are difficult results from the 1860s on, starting with Riemann's use of an odd non-degenerate (nonzero gradient at the Jacobian origin) Θ. He used this to express functions on any Riemann surface through there divisors: Generalizing Abel's Theorem. Unlike the genus 1 case, you can't analytically continue one odd Θ over the whole locus of genus g curves; it will become degenerate in large sub-loci.

Usually, however, it is the even Θ functions that express properties of a special locus. The start of such function theory is the evaluation of even nondegenerate Θs at the origins of the Jacobians. That gives the closest approximation for something automorphic along L. On, however, special loci L, you can only be certain that some of the 22g-1+2g-1 will be nonzero. For example, if the locus defines an especially useful Θ canonically, there is the problem of determining if the ϑ-null will be nonzero anywhere on L.

We often consider those L that have an attached canonical ϑ-null. For example, this happens if L is the image of covers of the sphere with odd-order branch cycles ([Fr09a, §6.1] or [Se90a, §1]; the covers don't have to have abelian monodromy group). Reason: These have an associated to a special half-canonical divisor class [Dl] as l varies along L, the check for that would be that at general points of L, the linear system of Dl contains no positive divisor [Fr09a, Prop. 6.12].

The sub-loci Riemann knew about best were those of hyperelliptic curves. The goal of this site is to encourage investigations of other subloci, especially those that are the images of useful Hurwitz spaces as they enter various Schottky problems, the Regular Inverse Galois problem, and the Modular Tower Program. This site welcomes the chance to advertise papers that contribute to the study of special loci L. 

Since I'm not at all expert on the classical literature, expositions that reach into that literature are welcome. Fay's book [Fa73] is a convenient reference source, neither too ancient nor too modern. It dedicates much analysis to expressions for various objects on Riemann surfaces of genus g near degenerate loci. That is, it deals with the boundary of the moduli of curves of genus g, in a style similar to that of, say, Poincaré. We are especially interested in papers that consider the behavior close to the cusps of Hurwitz spaces. 

[Fa73] J. Fay, Theta Functions on Riemann Surfaces, Lecture notes in Mathematics 352, Springer Verlag, Heidelberg, 1973.

[Fr09a] M. D. Fried,  Alternating groups and moduli space lifting Invariants, description and properties of spaces of 3-cycle covers, Arxiv #0611591. 01/04/09 To appear in Israel J.

[Se90a] J.P. Serre, Relèvements dans Ãn, C. R. Acad. Sci. Paris 311 (1990), 477–482.