arXiv subjects

Brett Nasserden

Publications and source records attributed to Brett Nasserden.

12 recordsLinked to original sources

arXiv2026

Finite extensions of abelian schemes and duality for commutative group stacks

Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model---that is, a proper flat finitely presented $R$-group scheme with abelian generic fibre---we prove a finer structure theorem: its normalization is an abelian scheme, and the model is obtained from it by a pushout involving finite flat group schemes. An analogous abelian quotient exists for proper flat commutative group stacks with finite flat inertia. These structure theorems give explicit quotient presentations for duals. Over an arbitrary base, we show that the dual of a proper flat finitely presented commutative group algebraic space is algebraic, proper, flat, and finitely presented, and biduality holds. When $2$ is invertible, the corresponding result for group stacks proves Brochard's duality conjecture. As an application over a discrete valuation ring, Polishchuk's kernel-algebra Fourier--Mukai argument gives equivalences between the unbounded derived categories of quasi-coherent sheaves on every proper flat finitely presented commutative group scheme and its dual, and between their bounded derived categories of coherent sheaves, with no tameness or residue-characteristic hypothesis. For a quasiabelian model, the dual category is a finite-flat equivariant category on its abelian normalization.

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arXiv2026

Vertex operator algebra bundles on modular curves and their associated modular forms

This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to $V$-valued quasi-modular forms. The space $Q(V)$ of $V$-valued quasi-modular forms is endowed with the structure of a doubled QVOA, and in particular the algebra $Q$ of quasi-modular forms is itself a doubled QVOA. $Q(V)$ also admits a natural derivative operator arising from the connection on the bundle defined by $V$ and the modular derivative, which we call the raising operator. We introduce an associated lowering operator $\Lambda$ on $Q(V)$ having the property that the $V$-valued modular forms $M(V)\subseteq Q(V)$ are the kernel of $\Lambda$. This extends the classical theory of scalar-valued quasi-modular forms. We exhibit an explicit isomorphism of $M(V)$ with $M \otimes V$. Finally, the coordinate invariance of vertex operators implies that $M(V)$ has a natural Hecke theory, and we use this isomorphism to fully describe the Hecke eigensystems: they are the same as the systems of eigenvalues that arise from scalar-valued quasi-modular forms.

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arXiv2025

Dynamics of Projectivized Toric Vector Bundles

We study surjective endomorphisms of projective bundles over toric varieties, achieving three main results. First, we provide a structural theorem describing endomorphisms of projectivized split bundles over arbitrary base varieties, which we use to classify all surjective endomorphisms of Hirzebruch surfaces and construct novel families of examples. Second, for non-split equivariant bundles over toric varieties, we prove that the dynamical degree of an endomorphism of the projectivization is controlled by the base morphism; as a consequence, we establish the Kawaguchi--Silverman conjecture for such bundles. Third, using an explicit transition function method, we prove that projectivizations of tangent and cotangent bundles of smooth toric varieties admit no non-automorphic surjective endomorphisms commuting with toric morphisms on the base.

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arXiv2023

Mukai Duality for abelian stacks

An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\mathcal{A}$ has a dual $\mathfrak{D}(\mathcal{A})$ which generalizes the classical dual. In general, $\mathfrak{D}(\mathcal{A})$ is no longer an abelian stack but a commutative group scheme which is an extension of a finite, flat, and finitely presented commutative group scheme by an abelian scheme. We show that Fourier-Mukai duality holds for tame abelian stacks and their duals. Our approach is as follows. Let $QC_\infty(\mathcal{A})$ be the stable infinity category of quasi-coherent sheaves on $\mathcal{A}$. We define a Poincare bundle on $\mathcal{A}\times \mathfrak{D}(\mathcal{A})$ and use this to show that $QC_\infty(\mathcal{A})$ and $QC_\infty(\mathfrak{D}(\mathcal{A}))$ are dual as objects in the infinity category of stable infinity categories. By a result of Ben-Zvi,Francis and Nadler we have that $QC_\infty(\mathcal{A})$ is self dual, giving that $QC_\infty(\mathcal{A})\cong QC_\infty(\mathfrak{D}(\mathcal{A}))$ which gives the statement for the derived categories. In addition we give new examples of tame abelian stacks.

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arXiv2023

Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective bundles on elliptic curves, thereby completing the conjecture for all projective bundles on curves. Our approach is to use the transition functions of the bundles. This allows us to further prove the conjecture for projective split bundles on a smooth projective variety with finitely generated Mori cone.

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arXiv2022

On the realizability of arithmetic degrees of morphisms

The Kawaguchi-Silverman conjecture relates two different invariants of a surjective endomorphism, the dynamical and arithmetic degrees. As the Kawaguchi-Silverman conjecture is only meaningful when a morphism has a Zariski dense orbit, it has no content for varieties with positive Kodaira dimension. A generalization of the Kawaguchi-Silverman conjecture which is meaningful in positive Kodaira dimension is the so called sAND conjecture, which involves the set of "small" arithmetic degrees. Kawaguchi and Silverman showed that a small arithmetic degree is the modulus of an eigenvalue of $f^*\colon N^1(X)\rightarrow N^1(X)$. In this article we investigate which possible eigenvalues arise as an arithmetic degree. We show that surjective endomorphisms of abelian varieties may have eigenvalues which are not arithmetic degrees. Conversely, we show that every eigenvalue of a surjective endomorphism of a toric variety is an arithmetic degree using the minimal model program. Finally, we investigate how the minimal model program may be applied to study this realizability question for varieties that admit an int-amplified endomorphism.

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arXiv2022

Some applications of the minimal model program in arithmetic dynamics

We describe a general program for studying the dynamics of surjective endomorphisms of algebraic varieties that are amenable to techniques from the minimal model program. We obtain density results on the pre-periodic points of surjective endomorphisms of varieties admitting an int-amplified endomorphism, and reduce certain cases of the Medvedev-Scanlon conjecture to so called Q-abelian varieties using our approach. We also provide a connection between the existence of an automorphism with positive entropy and group of connected components of a variety. In particular, we show that if $X$ is normal and projective with finitely generated nef cone then $X$ has an automorphism of positive entropy if and only if the group of connected components $π_0\Aut(X)$ has an element of infinite order.

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arXiv2021

Heights and quantitative arithmetic on stacky curves

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space $\mathbb{P}^1$, is equivalent to the $abc$-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

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arXiv2020

Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

Let $f\colon X\rightarrow X$ be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of $f$ may be studied through the dynamics of the linear action $f^*\colon Pic(X)_\mathbb{R}\rightarrow Pic(X)_\mathbb{R}$, which are governed by the spectral theory of $f^*$. Let $λ_1(f)$ be the spectral radius of $f^*$. We study $\mathbb{Q}$-divisors $D$ with $f^*D=λ_1(f) D$ and $κ(D)=0$ where $κ(D)$ is the Iitaka dimension of the divisor $D$. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. Finally we identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture.

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arXiv2020

The density of rational points on $\mathbb{P}^1$ with three stacky points

In this paper we consider the density of rational points on the "stacky" curve $\mathcal{X}(\mathbb{P}^1;0,2;1,2;\infty,2)$ which is $\mathbb{P}^1$ with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown height. In particular, we prove a conjecture of Ellenberg.

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arXiv2019

Uniformity of fibres of period mappings and the $S$-unit equation

In this paper we give a refinement of the method introduced by Lawrence and Venkatesh and thereby showing that their proof of Mordell's conjecture is uniform up to a uniform bound on the number of Galois representations attached to some family of abelian varieties. We are also able to give an unconditional proof of a uniform boundedness statement on the number of solutions to $S$-unit equations, which qualitatively is best possible, recovering a result of Evertse.

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arXiv2018

Arithmetic aspects of the Burkhardt quartic threefold

We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for the universal genus 2 curve over it, as a double cover of the projective line. We show that the j-planes in the Burkhardt quartic mark the order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse pencil on a j-plane gives rise to the universal curve as a discriminant of a cubic genus one cover.

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