Computing Periods of Hypersurfaces

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.Comment: 33 pages; Final version. Fixed typos, minor expository changes. Changed code repository lin

Prym varieties of genus four curves

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 4.1

A compactification of the moduli space of multiple-spin curves

We construct a smooth Deligne–Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m, we give a combinatorial description of the local structure of the corresponding coarse moduli spaces. We also classify all irreducible and connected components of the resulting moduli spaces of multiple-spin curves

On reconstructing subvarieties from their periods

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 adde

An Octanomial Model for Cubic Surfaces

We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms, written in moduli from the E6 hyperplane arrangement. If such a surface is tropically smooth then its 27 tropical lines are distinct. We focus on explicit computations, both symbolic and p-adic numerical.Comment: 20 pages; clarified exposition at key points; final versio

Deep Learning Gauss–Manin Connections

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 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 pencils 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 lattices and the endomorphism fields of their transcendental lattices. © 2022, The Author(s)

Deep Learning Gauss-Manin Connections

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 page

Separation of periods of quartic surfaces

The periods of a quartic surface X are complex numbers obtained by integrating a holomorphic 2-form over 2-cycles in X. For any quartic X defined over the algebraic numbers and any 2-cycle G in X, we give a computable positive number c(G) such that the associated period A satisfies either |A| > c(G) or A = 0. This makes it possible in principle to certify a part of the numerical computation of the Picard group of quartics and to study the Diophantine properties of periods of quartics

Heights on curves and limits of Hodge structures

We exhibit a precise connection between Néron–Tate heights on smooth curves and biextension heights of limit mixed Hodge structures associated to smoothing deformations of singular quotient curves. Our approach suggests a new way to compute Beilinson–Bloch heights in higher dimensions

Enumerative geometry of double spin curves

Diese Dissertation hat zwei Teile. Im ersten Teil untersuchen wir die Modulräume von Kurven mit multiplen Spinstrukturen. Wir stellen eine neue Kompaktifizierung dieser Räume mit geometrisch sinnvollem Grenzverhalten vor. Die irreduziblen Komponenten dieser Räume werden vollstandig klassifiziert. Die Ergebnisse aus diesem ersten Teil der Dissertation sind fundamental für die Degenerationstechniken im zweiten Teil. Im zweiten Teil untersuchen wir eine Reihe von Problemen, die von der klassischen Geometrie inspiriert werden. Unser Hauptaugenmerk liegt hierbei auf dem Fall von zwei Hyperebenen, die eine kanonische Kurve in jedem Schnittpunkt tangential berühren. Wir fragen, ob eingemensamer Tangentialpunk existieren kann. Unsere Analyse zeigt, dass so ein gemeinsamer Punkt nur in Kodimension 1 im Modulraum existieren kann. Wir berechen dann weiter die Klasse dieses Divisors. Insbesonders zeigen wir, dass diese Klasse eine hinreichend kleine Steigung hat, sodass die kanonischen Klassen von Modulräumen von Kurven mit zwei ungeraden Spinstrukturen gross ist, wenn der Genus grösser ist als neun. Falls die zugehörigen groben Modulräume gutartige Singularitäten haben, dann haben sie in diesem Intervall maximale Kodaria Dimension.This thesis has two parts. In Part I we consider the moduli spaces of curves with multiple spin structures and provide a compactification using geometrically meaningful limiting objects. We later give a complete classification of the irreducible components of these spaces. The moduli spaces built in this part provide the basis for the degeneration techniques required in the second part. In the second part we consider a series of problems inspired by projective geometry. Given two hyperplanes tangential to a canonical curve at every point of intersection, we ask if there can be a common point of tangency. We show that such a common point can appear only in codimension 1 in moduli and proceed to compute the class of this divisor. We then study the general properties of curves in this divisor. Our divisor class has small enough slope to imply that the canonical class of the moduli space of curves with two odd spin structures is big when the genus is greater than 9. If the corresponding coarse moduli spaces have mild enough singularities, then they have maximal Kodaira dimension in this range