The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the space.
Passing from the functions to their zero-sets fibers this set over the space of infinite discrete subsets of the disk, also some sort of infinite-dimensional complex manifold.
I'd like to know if there's a general obstruction to the existence of any kind of holomorphic cross-section. Perhaps even there are no non-constant holomorphic functions on the base space at all?
I understand that there's no naive generalization available from the usual elementary symmetric polynomials to infinitely many variables. Indeed the delicate nature of the Weierstrass factorization theorem reflects that. So I suppose I'm looking for a general principle that would imply that any Weierstrass-like theorem must involve, in an essential way, putting something like an order on the zero set, so that the output can't vary nicely across all zero-sets.
