Section: New Results

A numerical transcendental method in algebraic geometry

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. In [10], Pierre Lairez and Emre Sertöz focus on algebraic surfaces and give a numerical method to compute Picard groups. As an application, they count smooth rational curves on quartic surfaces using the Picard group. It is the first time that this kind of computation is performed.