Twelve Points on the Projective Line, Branched Covers, and Rational Elliptic Fibrations

The following divisors in the space Sym12 P1 of twelve points on P 1 are actually the same: (A) The possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P1 ; (E) the branch locus of a degree 3 map from a genus 4 curve to P 1 induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections between these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if C → P1 is a cover as in (D), then there are 135 different such covers branched at the same points; (iii) if p ∈ P2 ⊃ C is a configuration as described in (C), then there are 120 essentially different such configurations with the same tangent lines. Some parts of this story are well-known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas’ trigonal construction and Shioda’s E8-Mordell-Weil lattice.