Constructing Clifford quantum P3s with finitely many points

We present an algebro-geometric technique for constructing regular Clifford algebras A of global dimension four with associated point scheme consisting of a prespecified finite number of points. In particular, if A has more than one point in its point scheme, then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors; these cubic divisors correspond to regular Clifford subalgebras of A of global dimension three. If A has exactly a finite number, n, of distinct points in its point scheme, then n∈{1,3,4,5,…,13,14,16,18,20} and all these possibilities occur. We also prove that if a regular Clifford algebra R of global dimension d⩾2 has exactly a finite number, n, of distinct isomorphism classes of point modules, then n is odd if and only if R is an Ore extension of a regular Clifford subalgebra of R of global dimension d−1.