A Numerical Transcendental Method in Algebraic Geometry: Computation of Picard Groups and Related Invariants

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces in $\mathbb{P}^3$. As an application, we count the number of rational curves of a given degree lying on each surface. For quartic surfaces we also compute the endomorphism ring of their transcendental lattice. The method applies more generally to the computation of the lattice generated by Hodge cycles of middle dimension on smooth projective hypersurfaces. We demonstrate the method by a systematic study of thousands of quartic surfaces (K3 surfaces) defined by sparse polynomials. The results are only supported by strong numerical evidence, yet the possibility of error is quantified in intrinsic terms, like the degree of curves generating the Picard group.