arXiv subjects

Eoin Mackall

Publications and source records attributed to Eoin Mackall.

12 recordsLinked to original sources

arXiv2026

Refined index obstructions for Brauer classes on an abelian variety

We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions are more stringent than previous obstructions and, as a consequence, we produce more counterexamples to the integral Hodge conjecture. Throughout this work, we focus on algorithmic aspects of these obstructions and we illustrate many of these aspects through the concrete examples of complex abelian varieties.

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arXiv2026

An algorithmic reduction to canonical forms for vector bundles on anisotropic conics

We describe a polynomial complexity algorithm for reducing transition matrices, for vector bundles glued along a clutching-type cover of a real anisotropic conic, to canonical block diagonal forms. This is a generalization, to the real anisotropic form, of the classification of vector bundles on the Riemann sphere by their canonical diagonal forms due to Grothendieck and Birkhoff. To enable our algorithm, we provide an elementary algebraic proof for the result, due to Biswas-Nagaraj and Novakovic, of the decomposition of vector bundles on real anisotropic conics into sums of indecomposable vector bundles of rank at most 2. While our algorithm and our proof of this decomposition focus solely on the setting of a real anisotropic conic, our methods are immediately generalizable to anisotropic conics over arbitrary fields.

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arXiv2025

A Looming of phantoms

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of the second Hirzebruch surface at 9 points in general position. We also pose conjectures about the existence of phantom subcategories in the derived categories of other rational surfaces, obtained as the blowups of the other Hirzebruch surfaces.

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arXiv2025

On deformations of Azumaya algebras with quadratic pair

We construct a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair. Under the assumption that either 2 is a global unit or the algebra is of degree 2, we show how the deformation theory of these objects reduces to the deformation theory of the underlying Azumaya algebra. Namely, if the underlying Azumaya algebra has unobstructed deformations then so does the quadratic pair. On the other hand, in the purely characteristic 2 setting, we construct an Azumaya algebra with unobstructed deformations which can be equipped with a quadratic pair such that the associated triple has obstructed deformations. Our example is a biquaternion Azumaya algebra on an Igusa surface. Independently from the above results, we also introduce a new obstruction for quadratic pairs, existing only in characteristic 2, which is intermediate to both the strong and weak obstructions that were recently introduced by Gille, Neher, and the second named author. This intermediate obstruction characterizes when a canonical extension of the Lie algebra sheaf of the automorphism group scheme of some quadratic triple is split.

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arXiv2024

On the algebraizability of formal deformations in $K$-cohomology

We show that algebraizability of the functors $R^1\pi_*\mathcal{K}^M_{2,X}$ and $R^2\pi_*\mathcal{K}^M_{2,X}$ is a stable birational invariant for smooth and proper varieties $\pi:X\rightarrow k$ defined over an algebraic extension $k$ of $\mathbb{Q}$. The same is true for the \'etale sheafifications of these functors as well. To get these results we introduce a notion of relative $K$-homology for schemes of finite type over a finite dimensional, Noetherian, excellent base scheme over a field. We include this material in an appendix.

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arXiv2024

Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties

We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically closed field $k$ and if $F/k$ is a perfect field extension, then there exists a principal homogeneous space for $E_F$ splitting a Severi--Brauer variety $X=\mathrm{SB}(A)$ over $F$ if and only if $A$ is Brauer equivalent to a cyclic algebra. Along the way, we also give a uniform proof of a generalization of results due to O'Neil, Clark and Sharif, and Antieau and Auel. Second, we give an example of an elliptic curve $E$ over a field $k$ together with a central simple algebra $A/k$ of degree $4$ such that $E$ is the Jacobian of a smooth genus one curve $C$ embedded in the Severi--Brauer variety $X=\mathrm{SB}(A)$ as a degree 8 curve and such that $E$ is not the Jacobian of any genus one curve of smaller degree contained in $X$. Our example is, in some sense, as small as possible in both dimension and arithmetic complexity. Third, we show that for any odd integer $n\geq 3$ there is a central simple algebra $A$ of degree $n$ over a local field $k$ which is split by the principal homogeneous space of an elliptic curve $E/k$ but not by any principal homogeneous space for any quadratic twist of $E$. This generalizes a recent result of Saltman in the case of surfaces to arbitrary even dimension.

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arXiv2023

Pro-representability of $K^M$-cohomology in weight 3 generalizing a result of Bloch

We generalize a result, on the pro-representability of Milnor $K$-cohomology groups at the identity, that's due to Bloch. In particular, we prove, for $X$ a smooth, proper, and geometrically connected variety defined over an algebraic field extension $k/\mathbb{Q}$, that the functor \[\mathscr{T}_{X}^{i,3}(A)=\ker\left(H^i(X_A,\mathcal{K}_{3,X_A}^M)\rightarrow H^i(X,\mathcal{K}_{3,X}^M)\right),\] defined on Artin local $k$-algebras $(A,\mathfrak{m}_A)$ with $A/\mathfrak{m}_A\cong k$, is pro-representable provided that certain Hodge numbers of $X$ vanish.

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arXiv2022

On the Chow groups of a biquaternion Severi--Brauer variety

We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.

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arXiv2021

The Period and Index of a Generic Geometrically Elliptic Normal Curve

We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of these curves showing that all possible period/index combinations are possible.

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arXiv2021

$\mathbb{A}^1$-homotopy equivalences and a theorem of Whitehead

We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive $\mathbb{A}^1$-homotopy equivalences between smooth projective varieties.

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arXiv2020

Universal additive Chern classes and a GRR-type theorem

We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor associates to $X$ is related to the associated graded ring of the $γ$-filtration on the Grothendieck ring of finite rank locally free sheaves on $X$ via a Grothendieck-Riemann-Roch type theorem.

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arXiv2016

Six variations on a theme: almost planar graphs

A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\forall v$, $G-v$ planar; $\forall e$, $G-e$ planar; and $\forall e$, $G/e$ planar. The Graph Minor Theorem of Robertson and Seymour ensures that each of these six gives rise to a finite set of obstruction graphs. For the three definitions with universal quantifiers we determine this set. For the remaining properties, apex, edge apex, and contraction apex, we show there are at least 36, 55, and 82 obstruction graphs respectively. We give two similar approaches to almost nonplanar ($\exists e$, $G+e$ is nonplanar and $\forall e$, $G+e$ is nonplanar) and determine the corresponding minor minimal graphs.

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