w/ E. Sertöz
In “A transcendental method in algebraic geometry”, Griffiths emphasized the role of certain multivariate integrals, known as periods, “to construct a continuous invariant of arbitrary smooth projective varieties”. Periods often determine the projective variety completely and therefore its algebraic invariants. Translating periods into discrete algebraic invariants is a difficult problem, exemplified by the long standing Hodge conjecture which describes how periods determine the algebraic cycles within a projective variety.
Recent progress in computer algebra makes it possible to compute periods with high precision and put transcendental methods into practice. We focus mainly on algebraic surfaces and give a numerical method to compute Picard groups. As an application, we count smooth rational curves on quartic surfaces using the Picard group.
A K3 surface containing 133056 smooth rational curves of degree 4 generating a Picard group of rank 19.