We develop algebraic geometry for coupled cluster theory using second quantization. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The truncated eigenstates parametrize well known varieties such as the Grassmannian, flag varieties and spinor varieties. We will offer a detailed study into the truncation varieties. Additionally in second quantization we work within the exterior algebra. There we can define interior and exterior operators called the creation and annihilation operators. They span a special Clifford algebra called the Fermi-Dirac algebra, which we will study as well.