On reconstructing subvarieties from their periods

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.

arXiv comment16 pages; computational aspects highlighted; reconstruction of twisted cubics in higher degree surfaces added

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Cited by in corpus (3)

• Deep Learning Gauss-Manin Connections
• Effective obstruction to lifting Tate classes from positive characteristic
• Toric differential forms and periods of complete intersections