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arXiv · 1908.03221

On reconstructing subvarieties from their periods

Hossein Movasati·Emre Can Sertöz
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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.

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Hossein Movasati, Emre Can Sertöz. 2019-08-08. On reconstructing subvarieties from their periods. https://doi.org/10.1007/s12215-020-00562-x

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