Computing examples of Hurwitz correspondences in cyclic and non-cyclic portraits

A Hurwitz correspondence is a multi-valued self-map of the moduli space of genus zero curves with n marked points, M0,n. Considering two n-marked curves (C, a1, ..., an) and (D, b1, ..., bn), given a map ρ from {1, ..., n} to itself and some ramification data, one obtains a Hurwitz correspondence on M0,n which sends a marked curve (D, b1, ..., bn) to any of the marked curves (C, a1, ..., an) for which we have a rational map φ : C → D with ai → bρ(i) that satisfies the required ramification data. The dynamical degree of a Hurwitz correspondence h is the maximal eigenvalue of the induced map (h)∗ on homology. We study the case of the induced map on the divisor class group with a particular focus on ramification types, as studied by Koch and Roeder. This experimental study considers the eigenvalues for several Hurwitz correspondences for maps φ of low degree. Our main experimental findings are that ∀λi, where λi is not a dynamical degree but is an eigenvalue of the induced map on homology (h)∗, then 1 ≤ |λi|≤ k, where k is the global degree of the rational map φ. Our results also suggest that when k < λmax, where λmax is the dynamical degree for a given induced map (h)∗ and k is the global degree of the rational map φ, then the multiplicity of λmax is 1. 1 Definitions and background First, we will begin by defining n-pointed smooth rational curves and the moduli space of curves. Definition 1. An n-pointed rational curve, (C, p1, ..., pn), is a projective rational curve C equipped with a choice of n distinct points, called the marked points. Definition 2. For n ≥ 3, there is a fine moduli space, M0,n, for the problem of classifying n-pointed smooth rational curves up to isomorphism. Example 1. When n = 3, there is a unique isomorphism from any smooth rational curve with three marked points (C, p1, p2, p3) to (P, 0, 1,∞). In this case M0,3 is a single point. For n = 4, every rational curve with four distinct marked points (C, p1, p2, p3, p4)is isomorphic to (P, 0, 1,∞, q) for some q ∈ P \ {0, 1,∞}. This shows that M0,4 is isomorphic to P \ {0, 1,∞}.