Anomaly cancellation for a $U(1)$ factor

We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $\mathfrak{u}_1$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $\mathfrak{u}_1$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Koll{\'a}r) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $\mathfrak{su}_3\oplus\mathfrak{su}_2$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $\mathfrak{u}_1$ hypercharges. We also discuss whether it is possible to find all solutions in this way.