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Remy van Dobben de Bruyn

Publications and source records attributed to Remy van Dobben de Bruyn.

11 recordsLinked to original sources
arXiv2026

Mermin-Peres magic rectangles modulo odd primes

The Mermin-Peres magic square provides a simple example of a system of linear equations over $\mathbb{Z}/2\mathbb{Z}$ which has no classical solutions but does have a finite-dimensional operator solution. For a long time, it was not known how to construct similar examples over $\mathbb{Z}/d\mathbb{Z}$ with $d$ odd. In this paper, we construct, for every integer $d \geq 2$, a linear system over $\mathbb{Z}/d\mathbb{Z}$ that has a finite-dimensional operator solution but no classical solution. For an odd prime $p$, our operators act on two $p$-dimensional qudits and generate a finite $p$-group obtained by adjoining diagonal polynomial phase operators to the generalized Pauli group, thereby forming a natural analogue of the Mermin--Peres magic square. Classical inconsistency follows from an elementary linearity argument comparing assignments on abelian subgroups. Furthermore, we prove a sharp threshold for the required degree of the added polynomial phase operators: our construction uses polynomials up to degree $p - 1$, and we show that the group formed by polynomials up to degree $p - 2$ is noncontextual.

quant-ph↗
arXiv2026

Divisors in projective bundles over the projective line whose complement is affine space

Given a smooth projective variety $X$ of dimension $n$ and a closed subscheme $Z \subseteq X$, it is in general a difficult problem to determine whether $X \setminus Z$ is isomorphic to $\mathbf A^n$. In the case where $X$ is a projective bundle over $\mathbf P^1$ and the restriction of every irreducible component of $Z$ to every fibre is a hyperplane, we give a complete geometric characterisation in terms of the irreducible components of $Z$ and their intersections. In the appendix, we use this to obtain a coordinate-free explanation for the recent first counterexample to the Jacobian conjecture.

math.AG↗
arXiv2026

A condensed proof of the pro-étale and étale exodromy theorems

The exodromy correspondence of Barwick, Glasman, and Haine computes constructible sheaves of spaces on a scheme $X$ as an $\infty$-category of continuous functors from the profinite category $\operatorname{Gal}(X)$. Viewing $\operatorname{Gal}(X)$ instead as a condensed category, this was extended by Wolf to an exodromy correspondence for pro-étale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-étale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) étale sheaves that does not yet appear in the literature, which is closely related to Lurie's work on ultracategories. Finally, we use this to give a new proof of the constructible étale exodromy correspondence of Barwick, Glasman, and Haine. Without additional effort, our method removes the qcqs hypotheses on the schemes, and gives versions for sheaves with coefficients in more general $\infty$-categories. Finally, we refine the methods to obtain a $κ$-condensed statement for any uncountable cardinal $κ$ such that $κ> \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$.

math.AG↗
arXiv2024

Grothendieck Galois theory and étale exodromy

Finite étale covers of a connected scheme $X$ are parametrised by the étale fundamental group via the monodromy correspondence. This was generalised to an exodromy correspondence for constructible sheaves, first in the topological setting by MacPherson, Treumann, Lurie, and others, and recently also for étale and pro-étale sheaves by Barwick--Glasman--Haine and Wolf. The proof of the étale exodromy theorem is long and technical, using many new definitions and constructions in the $(\infty,2)$-category of $\infty$-topoi. This paper gives a quick proof of the étale exodromy theorem for constructible sheaves of sets (with respect to a fixed stratification), in the style of Grothendieck's Galois theory.

math.AG↗
arXiv2024

Constructible sheaves on toric varieties

This paper gives an explicit computation of the category of constructible sheaves on a toric variety (with respect to the stratification by torus orbits). Over the complex numbers, this simplifies a description due to Braden and Lunts. The same computation is carried out for split toric varieties over an arbitrary field, for constructible étale sheaves whose restriction to each stratum is locally constant and tamely ramified. This gives the first explicit computation of an étale exodromy theorem in positive characteristic for a stratification over a partially ordered set of height greater than 1.

math.AG↗
arXiv2020

The equivalence of several conjectures on independence of $\ell$

We consider several conjectures on the independence of $\ell$ of the étale cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main result is that independence of $\ell$ of the Betti numbers $h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of $\ell$ of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type $k$-schemes.

math.AG↗
arXiv2020

The Hodge ring of varieties in positive characteristic

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}(X) = \dim H^j(X,Ω_X^i)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the $h^{i,0}(X)$ and the $h^{0,j}(X)$ (and their duals $h^{i,n}(X)$ and $h^{n,j}(X)$). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

math.AG↗
arXiv2019

Automorphisms of categories of schemes

Given two schemes $S$ and $S'$, we prove that every equivalence between $\mathbf{Sch}_S$ and $\mathbf{Sch}_{S'}$ comes from a unique isomorphism between $S$ and $S'$. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki and fully answers a programme set out by Brandenburg in a series of questions on MathOverflow in 2011.

math.AG↗
arXiv2019

A variety that cannot be dominated by one that lifts

We prove a precise version of a theorem of Siu and Beauville on morphisms to higher genus curves, and use it to show that if a variety $X$ in characteristic $p$ lifts to characteristic $0$, then any morphism $X \to C$ to a curve of genus $g \geq 2$ can be lifted along. We use this to construct, for every prime $p$, a smooth projective surface $X$ over $\bar{\mathbb F}_p$ that cannot be rationally dominated by a smooth proper variety $Y$ that lifts to characteristic $0$.

math.AG↗