  Second-order superintegrable systems in dimensions two and three are
essentially classified. With increasing dimension, however, the non-linear
partial differential equations employed in current methods become unmanageable.
Here we propose a new, algebraic-geometric approach to the classification
problem - based on a proof that the classification space for irreducible
non-degenerate second-order superintegrable systems is naturally endowed with
the structure of a quasi-projective variety with a linear isometry action. On
constant curvature manifolds our approach leads to a single, simple and
explicit algebraic equation defining the variety classifying superintegrable
Hamiltonians that satisfy all relevant integrability conditions generically. In
particular, this includes all non-degenerate superintegrable systems known to
date and shows that our approach is manageable in arbitrary dimension. Our work
establishes the foundations for a complete classification of second-order
superintegrable systems in arbitrary dimension, derived from the geometry of
the classification space, with many potential applications to related
structures such as quadratic symmetry algebras and special functions.
