Robin de Jong

Second variation of Zhang's λ-invariant on the moduli space of curves

We compute the second variation of the λ-invariant, recently introduced by S. Zhang, on the complex moduli space Mg of curves of genus g ≥ 2, using work of N. Kawazumi. As a result we prove that (8g+4)λ is equal, up to a constant, to the β-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the λ-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.

Admissible constants for genus 2 curves

S.-W. Zhang recently introduced a new adelic invariant for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2.

Theta functions on the theta divisor.

We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to a description in terms of theta functions and their derivatives of the Weierstrass point locus on the associated Riemann surface.

Torus bundles and 2-forms on the universal family of Riemann surfaces

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e_1 induced by the Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary object called a_g occurs, which was discovered and studied by Kawazumi around 2008. We present alternative proofs of Kawazumis (unpublished) results on the second variation of a_g on M_g. Also we review some results that were previously obtained on the invariant a_g, with a focus on its connection with Faltingss delta-invariant and Hain-Reeds beta-invariant.

Normal functions and the height of Gross–Schoen cycles

We prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Symmetric roots and admissible pairing

Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant χ. We provide a simple geometric interpretation for this invariant χ, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant χ coincides with the invariant ϕ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus 2 case, and we obtain a new proof of the Bogomolov conjecture for curves of genus 2 over number fields.

Special values of canonical Green’s functions

We give a precise formula for the value of the canonical Green’s function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the ‘energy’ of the Weierstrass points in terms of a spectral invariant recently introduced by N. Kawazumi and S. Zhang. It follows that the energy is strictly larger than log 2. Our results generalize known formulas for elliptic curves.

Néron-Tate heights of cycles on jacobians

We develop a method to calculate the Neron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the Neron-Tate height of the difference surface, the Abel-Jacobi images of the square of the curve, and of any symmetric theta divisor. As applications we obtain a new effective positive lower bound for the essential minimum of any Abel-Jacobi image of the curve and a proof, in the case of jacobians, of a formula proposed by Autissier relating the Faltings height of a principally polarized abelian variety with the Neron-Tate height of a symmetric theta divisor.

Positivity of the Height Jump Divisor

We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight -1, pulled back along normal function sections. In the case of the normal function on M_g associated to the Ceresa cycle, our result proves a conjecture of Hain. As an application of our result we obtain that the Moriwaki divisor on \bar M_g has non-negative degree on all complete curves in \bar M_g not entirely contained in the locus of irreducible singular curves.

Canonical heights and division polynomials

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p -adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the ‘division polynomials’ associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.