arXiv subjects

Matthew Dawes

Publications and source records attributed to Matthew Dawes.

6 recordsLinked to original sources

arXiv2024

General type results for moduli of deformation generalised Kummer varieties

In arXiv:1710.01672, we obtained general type results for orthogonal modular varieties associated with moduli spaces of compact hyperk\"ahler manifolds of deformation generalised Kummer type (also known as 'deformation generalised Kummer varieties'). The orthogonal modular varieties were defined in terms of an integer 2d, corresponding to the degree of polarisation of the associated hyperk\"ahler manifolds. In arXiv:1710.01672, we showed these orthogonal modular varieties are of general type when 2d is square-free and sufficiently large. Here we show that the square-free condition can be removed.

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arXiv2022

Orbits in Lattices

We exhibit algorithms for calculating Tits' buildings and orbits of vectors in a lattice $L$ for certain subgroups of $\operatorname{O}(L)$. We discuss how these algorithms can be applied to understand the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.

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arXiv2021

The Baily-Borel compactification of a family of orthogonal modular varieties

We study the Baily-Borel compactification of a family of four-dimensional orthogonal modular varieties arising in connection with moduli and periods of compact hyperkähler manifolds of deformation generalised Kummer type. Our main results concern the classification of boundary components, their incidence relations and combinatorics.

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arXiv2019

A very short proof of the Borisov-Nuer conjecture

We prove a conjecture of Borisov and Nuer, which states that every element in the even unimodular lattice of signature (1,9) can be expressed as the difference of two elements of squared length -2. As a consequence, every unnodal Enriques surface admits an Ulrich line bundle.

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arXiv2018

Boundary combinatorics of orthogonal modular 4-folds

We study combinatorial problems related to the singularities and boundary components of toroidal compactifications of orthogonal modular varieties. In particular, those associated with the moduli of algebraic deformation generalised Kummer 4-folds.

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arXiv2017

On the Kodaira Dimension of the Moduli of Deformation Generalised Kummer Varieties

We prove general type results for orthogonal modular varieties associated with the moduli of compact hyperk\"ahler manifolds of deformation generalised Kummer type ('deformation generalised Kummer varieties'). In particular, we consider moduli spaces of deformation generalised Kummer fourfolds with split-polarisation of degree $2d$. Our main result is that when $d$ is prime or $2d$ is square-free then the associated modular varieties are of general type when $d$ exceeds bounds we determine, subject to the existence of certain low-weight cusp forms for $\operatorname{O}(2,n)$. As a corollary, we conclude that the corresponding moduli spaces are also of general type.

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