Determining rational functions by their critical points

Question

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, 1983), we know that there are only a finite number of rational functions $\phi(z)$ of degree $d$ with complex coefficients that are ramified exactly at the points $P_i$, up to postcomposition by invertible rational functions. (The latter preserve the ramification points, so they must be taken into account.) Suppose that $\phi_1, \ldots, \phi_n$ are a maximal collection of such functions with the property that $\phi_i = \sigma \circ \phi_j$ for some $\sigma \in \mathrm{Aut}(\mathbb{P}^1)$ implies $i = j$.

Is there an invariant that allows one to distinguish between $\phi_1, \ldots, \phi_n$?

If $P_1, \ldots, P_{2d-2}$ are all distinct, then Goldberg (Advances in Math, 1991) showed that the number of postcomposition classes of rational functions ramified at exactly these points is positive and bounded above by the Catalan number $\rho(d) = \frac{1}{d}\binom{2d-2}{d-1}$. Moreover, if the $P_i$ are in general position, then the number of classes is exactly $\rho(d)$.

For example, when $d = 3$, we may normalize the ramified points to be at $0, 1, \infty$, and $c$, and we may further assume that $0, 1$, and $\infty$ are fixed after postcomposing by a suitable automorphism. Then Goldberg's result says there are precisely 2 rational functions of this shape for a general choice of $c \in \mathbb{C}$. How do we distinguish between them?

One can work out this example explicitly to see that $$ \phi(z) = \frac{\alpha z^3 + (1-2\alpha)z^2}{(2-\alpha) z - 1}.$$ The fourth critical point is $c = \frac{2\alpha - 1}{\alpha(2-\alpha)}$, so we require that $\alpha \neq 0, \pm 1,2^{\pm 1}$ in order to have four distinct critical points. A given $c$ generically determines two values of $\alpha$. But a priori, what data can I specify in order to nail down one or the other of these two functions?

Answer 1

One answer is that there are NO rational expressions in $P_1, \dots, P_{2d-2}$ that allow you to single out one of the functions $\phi_i$, or that even allow you to single out a nonempty, proper subset of the set of all $\rho(d)$ functions. The proof of this is the usual argument of this type (also in Harris's paper on Galois groups of enumerative problems). The parameter space of degree $d$, $1$ dimensional linear systems on $\mathbb{P}^1$ is irreducible; it is just the Grassmannian of $2$-dimensional linear subspaces of $H^0(\mathbb{P}^1,\mathcal{O}(d))$. There is a dense, open subset $U$ parameterizing base point free linear systems with $2d-2$ distinct ramification points. As a dense open in an irreducible variety, also $U$ is irreducible. There is a morphism $f:U\to \text{Sym}^{2d-2}(\mathbb{P}^1)$, i.e., $f:U\to \mathbb{P}^{2d-2}$, sending such a linear system to the corresponding ramification divisor.

A rational expression in $P_1,\dots,P_{2d-2}$ is the same thing as a rational transformation defined on a dense open subset of $\mathbb{P}^{2d-2}$. So a rational expression choosing one $\phi_i$ would be the same thing as a rational section $s$ of $f$. Since $f$ is quasi-finite, the image of $s$ would be an irreducible component of $U$. As $U$ is irreducible, there can be no such rational section. More generally, a rational expression singling out a nonempty proper subset of size $\sigma$ is the same thing as a union of irreducible components of the domain which has degree $\sigma$ over the target. Once again, since $U$ is irreducible, this can only occur when $\sigma$ equals $\rho(d)$.

Of course you did not specify that you want your rule to be a "rational expression" in $P_1,\dots,P_{2d-2}$. Since $f$ is finite, \'etale over a dense open subset of $\mathbb{P}^{2d-2}$, obviously there do exist \'etale local sections, i.e., there do locally exist "algebraic" functions which pick out some $\phi_i$. I have not heard of any nice expression for such (e.g., in terms of other special functions such as modular functions).

Answer 2

If all of the points $P_i$ lie on a circle $\gamma \subset \mathbb{CP}^1$, there is a beautiful description in

Eremenko and Gabrielov
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.
Ann. of Math. (2) 155 (2002), no. 1, 105–129.

Let $\phi: \mathbb{P}^1 \times \mathbb{P}^1$ be one of the rational maps. The circle $\gamma$ divides $\mathbb{P}^1$ into two hemispheres; call them $N$ and $S$. The image $\phi(\gamma)$ is contained in a circle; call it $C$. (This is NOT obvious.)

Consider $\phi^{-1}(C) \cap N$. (An earlier version of this answer wrote $\phi^{-1}(\phi(\gamma))$, but I want the inverse image of the whole circle $C$, which might be larger.) This is a collection of noncrossing arcs whose end points lie at the points $P_i$. Ermenko and Gabrielov show that there is precisely one rational function $\phi$ for each possible connectivity of these arcs. For example, there are $5$ possible ways to draw $3$ noncrossing arcs connecting $6$ points on the boundary of a disc, and there are $5$ degree $4$ maps with $6$ specified critical points.

I would be very interested in knowing a generalization of this result to the case where the points do not lie on a circle; I am fairly certain none is known.