The rank 2 classification problem I: scale invariant geometries

In this first of a series of three papers we outline an approach to classifying 4d $\mathcal{N}{=}2$ superconformal field theories at rank 2. The classification of allowed scale invariant $\mathcal{N}=2$ Coulomb branch geometries of dimension (or rank) greater than one is a famous open problem whose solution will greatly constrain the space of $\mathcal{N}{=}2$ superconformal field theories. At rank 2 the problem is equivalent to finding all possible genus 2 Seiberg-Witten curves and 1-forms satisfying a special K\"ahler condition. This is tractable because regular genus 2 Riemann surfaces can be uniformly described as binary-sextic plane curves, and the Seiberg-Witten curves are families of such curves varying meromorphically over the two-dimensional base. There are also solutions consisting of families of degenerate genus-2 Riemann surfaces given by a bouquet of two elliptic curves which are described by a different set of curves. In this paper we set up and carry out the analysis of the generic case, i.e., those whose typical fiber is a regular genus-2 Riemann surface with no extended automorphism, and find the complete answer for polynomial coefficients.