  We develop algebraic geometry for coupled cluster (CC) theory of quantum
many-body systems. The high-dimensional eigenvalue problems that encode the
electronic Schr\"odinger equation are approximated by a hierarchy of polynomial
systems at various levels of truncation. The exponential parametrization of the
eigenstates gives rise to truncation varieties. These generalize Grassmannians
in their Pl\"ucker embedding. We explain how to derive Hamiltonians, we offer a
detailed study of truncation varieties and their CC degrees, and we present the
state of the art in solving the CC equations.
