Papers

Siegel-Jacobi forms are analogues of modular forms on the universal abelian variety over the moduli stack of abelian varieties. We prove a conjecture of Kramer and disprove a theorem of Runge.

This sets up lots of intersection theory of b-divisors (divisors on blowups, closely related to the log Chow ring), used in the 'Rings of Siegel-Jacobi forms' paper just above.

You can find worked examples for the admcycles computations in this paper here.

This is the 'universal double ramification cycle' paper, which is the other half of the proof of Conjecture A of [JPPZ1].

We show that the torsion in the logarithmic Picard stack of Molcho-Wise gives a clean way to compactify spaces of roots of line bundles.

We prove that double-double ramification cycles are tautological, via proving that the log double ramification cycle is log tautological. Molcho and Ranganathan proved similar things at a similar time, though the proofs seem to me very different; they prove a general result comparing strict and total pullbacks of cycles under log blowups, we use the shape of the terms appearing in Pixton's formula.

We do a lot of deformation theory to understand the multiplicities of the components of the double ramification locus. This is basically one half of the proof of Conjecture A of [JPPZ1].

This solved a number of conjectures of Maino. My student Pim Spelier has now written a logarithmic version of this paper, and it's quite a bit shorter and more general.

Here we introduced the log double ramification cycle, and showed that it is in some ways better-behaved than the classical one.

We relate two different approaches to DR cycles, via log geometry and via Brill-Noether loci on compactified jacobians. Contains a comparison lemma that has been useful in a number of subsequent works.

This was where I first applied the ideas from 'A Néron model of the universal jacobian' to DR problems, constructing the k-pluricanonical DR for all k (see also Marcus-Wise).

A very general algorithm for computing Néron-Tate heights of points on jacobians, with a MAGMA implementation. Mostly written at a workshop in Baskerville Hall in the Wye Valley.

Proving Hain's conjecture on the positivity of the height jump (see also work of Brosnan-Pearlstein).

Here we proved an analogue of Hain's positivity conjecture mentioned just above, for families of abelian varieties. The result is less general than the one above, but we obtain more refined information on the shape of the functions involved.

We prove an even more special case of Hain's positivity conjecture, this time for families of jacobians of curves. The proof is combinatorial, eventually reducing to some statement about electrical networks. We give a new proof of Silverman and Tate's results on specialisation of heights in families of abelian varieties.

This paper discusses some consequences for rational torsion points of various theorems on Néron models. We conjecture a generalisation of Silverman and Tate's results on heights in families, which we show would imply the Torsion Conjecture for abelian varieties. We show that families admitting separated Néron models (and satisfying a technical positivity condition) satisfy this conjecture. Unfortunately, most families do not admit separated Néron models.

We prove a very special case of Hain's positivity conjecture (see above for more general version).

Naive heights are a useful computational tool for finding the rational points on curves and their jacobians. The key properties one needs are that the naive height is close to the Néron-Tate height, and that one can compute all the rational points of bounded naive height. Classically naive heights are defined in terms of projective embeddings of the jacobian. Here we propose a new approach via Arakelov intersection theory, requiring no projective embeddings.

The Néron-Tate height pairing is a quadratic form on the group of rational points on an abelian variety. Here we give an algorithm (with MAGMA implementation) to compute it on jacobians of hyperelliptic curves. Steffen Müller did similar work at the same time; the implementation currently used in MAGMA is his.

This largely supersedes my earlier paper 'Néron models of jacobians over base schemes of dimension greater than 1' in the same journal. We no longer assume Néron models are separated, but rather give a tropical criterion for separatedness, and in fact classify all separated group models.

This paper asks when the jacobian of the universal stable curve admits a separated Néron model (rarely), and constructs a universal base-change after which a separated Néron model does exist. Contains the germs of lots of ideas that have been useful to me in enumerative geometry.

When a family of abelian varieties has a separated Néron model, one has good control of torsion points. But this does not happen very often.

In large part initiating study of Néron models over higher dimensional bases. By now this has mostly been superseded, in particular by 'Models of Jacobians of curves' (with G. Orecchia, S. Molcho and T. Poiret). Marcus-Wise have greatly simplified and improved the results of the most technical section (Section 4: Classification of vertical Cartier divisors on certain complete local rings). Unfortunately in this paper we assume Néron models are separated; dropping this assumption gives a more flexible and general theory, which we explore in 'Models of Jacobians of curves'. Unlike in the case over a Dedekind scheme, in general the uniqueness part of the NMP does not imply separatedness.

Kontsevich and Manin computed the number of genus-0 curves of degree d through 3d-1 points. We investigate when these curves are defined over the rational numbers.

We show that, if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded.

Computing saturations of binomial ideals is a small industry, with interesting applications to statistical problems. We introduce a coarser approach, where we do not try to compute the saturation directly, but only bound how far a given ideal is from being saturated. This is often enough to solve the original statistical problem. We give some computational examples where our algorithms outperform the previous state-of-the-art.

The goal of this research is to understand life-cycles of mosquitos, with a view to understanding how changes in climate and land-use will affect future disease risk. Collecting data is quite hard, and the models have many parameters. We introduced the idea of formally differentiating the model with respect to its parameters, which allowed us to perform gradient descent using significantly less data than needed by other methods.

The goal here was to model the yield of grasslands under various climate conditions. We compared a pre-existing process-based dynamic model with a simple regression model. Generally, regression was better for identifying large-scale trends, the process-based model gave more precise local information.

This article is aimed at mathematicians working in the Netherlands, giving an overview of open access issues, and practical information for how to navigate them.

Now superseded (and corrected) by 'Quasi-compactness of Néron models, and an application to torsion points' and 'Torsion points and height jumping in higher-dimensional families of abelian varieties'.