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The Gauss-Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is computationally expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss-Manin connection of a pencil of hypersurfaces. As an application, we compute the periods of 96% of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard numbers and the endomorphism fields of their transcendental lattices., Comment: 30 pages

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis., 33 pages; Final version. Fixed typos, minor expository changes. Changed code repository link

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric techniques, addressing this problem. We also apply these methods to answer a question of W. Hackbusch on the non-closedness of site-independent cyclic matrix product states for infinitely many parameters., 18 pages

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., Comment: 16 pages; computational aspects highlighted; reconstruction of twisted cubics in higher degree surfaces added

International audience; Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces in P3. As an application, we count the number of rational curves of a given degree lying on each surface. For quartic surfaces we also compute the endomorphism ring of their transcendental lattice. The method applies more generally to the computation of the lattice generated by Hodge cycles of middle dimension on smooth projective hypersurfaces. We demonstratethe method by a systematic study of thousands of quartic surfaces (K3 surfaces) defined by sparse polynomials. The results are only supported by strong numerical evidence; yet, the possibility of error is quantified in intrinsic terms, like the degree of curves generating the Picard group.

Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X can be obtained by intersecting the dual of the corresponding Cayley cubic with the dual of the quadric containing C. We take this construction to its limit, studying all smooth degenerations and proving that the construction, with appropriate modifications, extends to the complement of a specific divisor in moduli. We work over an arbitrary field of characteristic different from two in order to facilitate arithmetic applications., Comment: 30 pages; Some expository changes; removed erroneous (old) Thm 4.11 and changed (old) Thm 4.23 into (new) Thm 4.17

Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4.