On reconstructing subvarieties from their periods
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On reconstructing subvarieties from their periods Hossein Movasati1 · Emre Can Sertöz2 Received: 24 April 2020 / Accepted: 25 September 2020 © The Author(s) 2020
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. Keywords Algebraic geometry · Transcendental methods · Hodge theory · Computational aspects Mathematics Subject Classification 32J25 · 14Q10 · 32G20
1 Introduction The Hodge conjecture asserts that on a smooth projective variety over ℂ , the ℚ-span of cohomology classes of algebraic cycles and of Hodge cycles coincide [11]. One difficulty of this conjecture lies in the general lack of techniques that can reconstruct algebraic cycles from their cohomology classes. In this article, we take a computational approach to this reconstruction problem and develop Algorithm 3.1. The highlight of this algorithm is its practicality. We have a computer implementation of the method that allows us to give rigorous Picard group computations, see Sect. 3.1. * Emre Can Sertöz [email protected] Hossein Movasati [email protected] 1
Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina,110, Rio de Janeiro, RJ 22460‑320, Brazil
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Max Planck Institute for Mathematics in the Sciences, MPI MiS, Inselstraße 22, Leipzig 04103, Germany
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H. Movasati, E. C. Sertöz
Our approach parallels that of [13] where it is proven that the periods of a general curve in a high degree surface in ℙ3ℂ are sufficient to reconstruct the equations of the curve. Because period computations are expensive in practice, our intended applications are towards low degree hypersurfaces where the periods give partial, but substantial, information. The computation of the Picard rank of quartic surfaces in Sect. 3 is an example. On another note, we expect that one can experiment with reconstructing Hodge cycles in hypersurfaces where the Hodge conjecture is not known, using [21, 24] and the arguments here.
1.1 Outline of the method For a smooth hypersurface X ⊂ ℙn+1 of degree d, Griffiths residues [15] establishes a conℂ and cohomology classes 𝜔p on X, nection between homogeneous polynomials p on ℙn+1 ℂ see Sect. 2.1 for a precise statement. Let Su be the space of degree u ∈ ℤ homogeneou
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