# On reconstructing subvarieties from their periods

## TL;DR

This paper introduces a practical method to reconstruct subvarieties of projective hypersurfaces by computing their periods, enabling the recovery of algebraic cycles and equations of subvarieties, with applications to Picard group calculations.

## Contribution

The paper presents a new computational approach for reconstructing subvarieties from periods, combining numerical and symbolic methods, applicable to hypersurfaces over algebraic numbers.

## Key findings

- Successfully computed generators of Picard groups of quartic surfaces.
- Verified Picard group computations despite non-extremal Picard numbers.
- Method effectively combines transcendental number computations with algebraic reconstruction.

## Abstract

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.