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.
Highlights
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
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
Explicit determination of algebraic cycles on the surface X gives a lower bound on its Picard group
Summary
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. We take a computational approach to this reconstruction problem and develop Algorithm 3.1. The highlight of this algorithm is its practicality. 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
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