A numerical transcendental method in algebraic geometry

TL;DR

This paper introduces a numerical transcendental method combining high precision period computation and lattice reduction to analyze algebraic surfaces, enabling the calculation of Picard groups, Hodge cycles, and rational curves.

Contribution

It presents a novel numerical approach for computing Picard groups and Hodge lattices of algebraic surfaces, extending to a systematic study of quartic K3 surfaces.

Findings

Successfully computed Picard groups for thousands of quartic surfaces.

Counted rational curves of specified degrees on each surface.

Determined endomorphism rings of transcendental lattices for quartic surfaces.

Abstract

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.