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Alexander Duncan

Publications and source records attributed to Alexander Duncan.

At least 19 recordsLinked to original sources
arXiv2025

Representations of finite subgroups of Cremona groups

The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful representation of that dimension over the same field. We find the exact value for rank 1 and 2 over all fields. We prove that the value is infinite for all fields of positive characteristic and rank greater than one. For many fields of characteristic 0, which include number fields and the complex field, we show that the value is finite for all ranks. Finally, for all fields of characteristic 0, we prove that the dimension is bounded below by a function that is exponential in the rank.

math.AG↗
arXiv2025

The numerical Amitsur group

The Amitsur subgroup of a variety with a group action measures the failure of the action to lift to the total spaces of its line bundles. We introduce the "numerical Amitsur group," which is an approximation of the ordinary Amitsur subgroup that can be computed using only the Euler-Poincar\'e characteristic on the Picard group. As an application, we find a uniform upper bound on the exponent of the Amitsur subgroup that depends only on the dimension and arithmetic genus of the variety and is independent of the group. Finally, we compute Amitsur subgroups of toric varieties using these ideas.

math.AG↗
arXiv2025

Classifying torsors of tori with Brauer groups

Using Mackey functors, we provide a general framework for classifying torsors of algebraic tori in terms of Brauer groups of finite field extensions of the base field. This generalizes Blunk's description of the tori associated to del Pezzo surfaces of degree 6 to all retract rational tori, essentially the largest class for which this is possible.

math.AG↗
arXiv2025

Sustainable Grid through Distributed Data Centers: Spinning AI Demand for Grid Stabilization and Optimization

We propose a disruptive paradigm to actively place and schedule TWhrs of parallel AI jobs strategically on the grid, at distributed, grid-aware high performance compute data centers (HPC) capable of using their massive power and energy load to stabilize the grid while reducing grid build-out requirements, maximizing use of renewable energy, and reducing Green House Gas (GHG) emissions. Our approach will enable the creation of new, value adding markets for spinning compute demand, providing market based incentives that will drive the joint optimization of energy and learning.

cs.DC↗
arXiv2023

Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks

A line bundle is immaculate if its cohomology vanishes in every dimension. We give a criterion for when a smooth toric Deligne-Mumford stack has infinitely many immaculate line bundles. This answers positively a question of Borisov and Wang. As a byproduct, we describe the asymptotic behaviour of the collection of immaculate line bundles.

math.AG↗
arXiv2023

A note on an effective bound for the gonality conjecture

The gonality conjecture, proved by Ein--Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus $g$ can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least 4g-3 would work in the gonality theorem. In this note, we improve the degree bound to 4g-4 with two exceptional cases.

math.AG↗
arXiv2020

Consequences of the existence of exceptional collections in arithmetic and rationality

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a geometrically rational, smooth, projective threefold over the the field of rational numbers that possesses a full \'etale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full \'etale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.

math.AG↗
arXiv2020

Separable algebras and coflasque resolutions

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories. Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Th\'el\`ene. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.

math.AG↗
arXiv2019

Birational self-maps of threefolds of (un)-bounded genus or gonality

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.

math.AG↗
arXiv2018

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

math.AG↗
arXiv2018

Automorphisms of cubic surfaces in positive characteristic

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.

math.AG↗
arXiv2018

Finite quasisimple groups acting on rationally connected threefolds

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: $\mathfrak{A}_5$, $\operatorname{PSL}_2(\mathbf{F}_7)$, $\mathfrak{A}_6$, $\operatorname{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, $\operatorname{PSp}_4(\mathbf{F}_3)$, $\operatorname{SL}_2(\mathbf{F}_{7})$, $2.\mathfrak{A}_5$, $2.\mathfrak{A}_6$, $3.\mathfrak{A}_6$ or $6.\mathfrak{A}_6$. All of these groups with a possible exception of $2.\mathfrak{A}_6$ and $6.\mathfrak{A}_6$ indeed act on some rationally connected threefolds.

math.AG↗
arXiv2017

Regular pairs of quadratic forms on odd-dimensional spaces in characteristic 2

We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective space over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an application, we show that every quartic del Pezzo surface over a perfect field of characteristic 2 has a canonical rational point and, thus, is unirational.

math.AG↗
arXiv2017

The toric Frobenius morphism and a conjecture of Orlov

We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov's Conjecture on derived dimension.

math.AG↗
arXiv2016

Equivariant unirationality of del Pezzo surfaces of degree 3 and 4

A variety X with an action of a finite group G is said to be G-unirational if there is a G-equivariant dominant rational map V -> X where V is a faithful linear representation of G. This generalizes the usual notion of unirationality. We determine when X is G-unirational for any complex del Pezzo surface X of degree at least 3.

math.AG↗
arXiv2016

Twisted forms of toric varieties

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M.~Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras

math.AG↗
arXiv2016

Fixed points of a finite subgroup of the plane Cremona group

We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a G-fixed point.

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