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

The rank 2 classification problem I: scale invariant geometries