A numerical transcendental method in algebraic geometry

Pierre Lairez (1) , Emre Can Sertöz (2)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 MPI-MiS - Max Planck Institute for Mathematics in the Sciences

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of numerical error in terms of an intrinsic measure of complexity of each surface. 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 (K3s) defined by sparse polynomials. 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.

Domaines: Géométrie algébrique [math.AG]

Pierre Lairez, Emre Can Sertöz. A numerical transcendental method in algebraic geometry. 2018. ⟨hal-01932147v1⟩