arXiv subjects

David Urbanik

Publications and source records attributed to David Urbanik.

17 recordsLinked to original sources

arXiv2026

Algebraic Hodge generic points are dense

Let $f: X \to S$ be a quasi-projective family of varieties defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. We show that the points of $S(\overline{\mathbb{Q}})$ that are Hodge generic for the variation of Hodge structures associated to $f$ are analytically dense in $S(\mathbb{C})$. In fact, in the spirit of the Grothendieck period conjecture and under a large monodromy assumption, we prove the density of the points of $S(\overline{\mathbb{Q}})$ where the periods of the fibre do not satisfy extra relations 'up to degree $\delta$'. As a by-product, we also establish new instances of the Mumford-Tate conjecture, beyond the realm of abelian motives. When the base $S$ is a curve, we provide quantitative estimates for points satisfying these properties. The main technical contribution is a new result on relations satisfied by solutions of $G$-operators, which relies on height estimates due to Bombieri and Andr\'e.

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arXiv2026

Intersections and the B\'ezout Range: Abelian Varieties

Given subvarieties $X, Y$ of a complex algebraic variety $S$ of complementary dimension, must they intersect? When $S$ is projective space, this is a consequence of the classical B\'ezout theorem, and an analogue for simple abelian varieties was established by Barth in 1968. Moreover, the moving lemma suggests that, after suitable translations, one may arrange for intersections of the expected dimension. In this work, we obtain variants for simple abelian varieties in the spirit of the completed Zilber--Pink philosophy. When $X$ and $Y$ have complementary dimension, we show that the intersections $X \cap [n]Y$ are zero-dimensional for all but finitely many integers $n$, and that these intersections collectively give rise to an analytically dense subset of $X$ as $n$ varies. We moreover control those $n$ for which $X \cap [n] Y$ has a positive dimensional component uniformly in $X, Y$ and $A$. When $\dim X + \dim Y < \dim A$, we show that $X \cap [n]Y = \varnothing$ for a set of integers $n$ of asymptotic density one, except in the presence of intersections at torsion points.

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arXiv2026

Galois Orbit Bounds for Surface Degenerations

Given a smooth proper family $g : X \to S$ of surfaces over a number field $K \subset \mathbb{C}$, with $S$ an irreducible curve and $\eta \in S$ its generic point, we consider the general problem of constraining the locus $\textrm{NL}(S)$ in $S(\overline{K})$ of points $s$ where the Picard rank of $X_{s}$ is larger than the generic Picard rank. Assuming that the local system $\mathbb{V} = R^{2} g_{*} \mathbb{Z}$ admits a non-trivial monodromy logarithm $N$ at infinity, we give a general condition under which certain points of $\textrm{NL}(S)$ of unexpectedly large Picard rank satisfy a ``Galois-orbit'' height bound. This leads to the following result of Zilber-Pink type: Let $g : X \to S$ be a one-parameter family of polarized K3 surfaces admitting a non-trivial limit mixed Hodge structure and such that $S(\mathbb{C})$ contains a Hodge-generic point. Then the locus in $S(\mathbb{C})$ where the Picard rank jumps by $3$ or more is finite. Our arguments include a new technique for ``spreading out'' formal geometry, a study of the rigid geometry of equicharacteristic zero semistable surface degenerations, and use the model-free Hyodo-Kato theory of Colmez-Nizio\l.

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arXiv2025

On the Complexity of Atypical Special Points

Given an integral variation of Hodge structure $\mathbb{V}$ on a complex algebraic variety $S$, polarized by some bilinear form $Q : \mathbb{V} \otimes \mathbb{V} \to \mathbb{Z}$, it is believed that the set $\mathcal{A}^{\textrm{iso}}_{0} \subset S(\mathbb{C})$ of isolated atypical special points associated to $(\mathbb{V}, Q)$ forms a finite set. Here we show that the number of such points $s$ is $O(Q(t_{s}, t_{s})^{\varepsilon})$ for any $\varepsilon > 0$, where $t_{s}$ is a minimal integral Hodge tensor defining $s$ (in an appropriate sense). This resolves a conjecture of Grimm and Monnee.

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arXiv2024

Degrees of Hodge Loci

We prove asymptotic estimates for the growth in the degree of the Hodge locus in terms of arithmetic properties of the integral vectors that define it. Our methods are general and apply to most variations of Hodge structures for which the Hodge locus is dense. As applications we give asymptotic formulas controlling the degrees of Noether-Lefschetz loci associated to smooth projective hypersurfaces in $\mathbb{P}^3$, and the degrees of subvarieties of the Torelli locus parameterizing Jacobians split up to isogeny.

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arXiv2024

Effective atypical intersections and applications to orbit closures

We propose a unifying setting for dealing with monodromically atypical intersections that goes beyond the usual Zilber-Pink conjecture. In particular we obtain a new proof of finiteness of the maximal atypical orbit closures in each stratum of translation surfaces $\Omega \mathcal{M}_g (\kappa)$, as given by Eskin, Filip, and Wright. We also describe a concrete algorithm, implementable in principle on a computer, which provably computes all maximal orbit closures which are 'atypical' in a sense described by Filip. The same methods also give a general algorithm for computing atypical special loci associated to systems of differential equations, and in particular give an effective and o-minimal free proof of the geometric Zilber-Pink conjecture for variations of mixed Hodge structures.

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arXiv2023

Arithmetic Deformation of Line Bundles

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers $\mathscr{O}_{L}[1/N]$ of a number field $L$, we produce a proper closed subscheme $\mathscr{E} \subsetneq \mathscr{S}$ outside of which all line bundles appearing in positive characteristic fibres of $f$ admit characteristic zero lifts. This in particular applies to elliptic surfaces over $\mathbb{P}^1$ and projective hypersurfaces in $\mathbb{P}^3$ of degree $d \geq 5$. We also study the locus in $\mathscr{E}$ in more detail in the $h^{0, 2} = 2$ case.

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arXiv2023

Algebraic Cycle Loci at the Integral Level

Let $f : X \to S$ be a smooth projective family defined over $\mathcal{O}_{K}[\mathcal{S}^{-1}]$, where $K \subset \mathbb{C}$ is a number field and $\mathcal{S}$ is a finite set of primes. For each prime $\mathfrak{p} \in \mathcal{O}_{K}[\mathcal{S}^{-1}]$ with residue field $κ(\mathfrak{p})$, we consider the algebraic loci in $S_{\overline{κ(\mathfrak{p})}}$ above which cohomological cycle conjectures predict the existence of non-trivial families of algebraic cycles, generalizing the Hodge loci of the generic fibre $S_{\overline{K}}$. We develop a technique for studying all such loci, together, at the integral level. As a consequence we give a non-Zariski density criterion for the union of non-trivial ordinary algebraic cycle loci in $S$. The criterion is quite general, depending only on the level of the Hodge flag in a fixed cohomological degree $w$ and the Zariski density of the associated geometric monodromy representation.

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arXiv2023

Existence and density of typical Hodge loci

Motivated by a question of Baldi-Klingler-Ullmo, we provide a general sufficient criterion for the existence and analytic density of typical Hodge loci associated to a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$. Our criterion reproves the existing results in the literature on density of Noether-Lefschetz loci. It also applies to understand Hodge loci of subvarieties of $\mathcal{A}_g$ . For instance, we prove that for $g \geq 4$, if a subvariety $S$ of $\mathcal{A}_g$ has dimension at least $g$ then it has an analytically dense typical Hodge locus. This applies for example to the Torelli locus of $\mathcal{A}_g$

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arXiv2023

Geometric G-functions and Atypicality

We describe a general method for giving $p$-adic interpretations of $G$-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to Andr\'e for bounding heights of moduli points where period functions acquire unusual algebraic relations. This leads to new results on Galois lower bounds for special moduli, and new cases of the Zilber-Pink conjecture. In particular, we establish the first Galois-orbit lower bounds on CM moduli in non-Shimura settings. As a more technical contribution, we introduce a refinement of the Pila-Zannier strategy capable of handling Zilber-Pink-type atypical intersection problems in arbitrary dimension and for arbitrary smooth projective families.

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arXiv2022

Unlikely Intersections with Bruhat Strata

Let $\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{Z}$, and let $\mathcal{T} \subset \mathcal{A}_{g}$ be a closed locus, also defined over $\mathbb{Z}$. Motivated by unlikely intersection conjectures, we study the intersection of $\mathcal{T}_{\mathbb{F}_{p}}$ with the Bruhat strata in $\mathcal{A}_{g,\mathbb{F}_{p}}$ as $p$-varies; these are strata characterized by the existence of certain subgroup schemes inside the $p$-torsion of the fibres. We find that, away from a finite set of primes, positive-dimensional ``unlikely'' intersections of $\mathcal{T}_{\mathbb{F}_{p}}$ with such strata are all accounted for by intersections of $\mathcal{T}$ with special loci inside $\mathcal{A}_{g}$. This result generalizes to all abelian-type Shimura varieties, and variations of Hodge structures equipped with certain motivic data. It moreover gives another example of how functional transcendence principles in characteristic zero can be used to study unlikely intersections in positive characteristic, building on recent work by the author.

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arXiv2021

Effective Methods for Diophantine Finiteness

Let $K \subset \mathbb{C}$ be a number field, and let $\mathcal{O}_{K,N} = \mathcal{O}_{K}[N^{-1}]$ be its ring of $N$-integers. Recently, Lawrence and Venkatesh proposed a general strategy for proving the Shafarevich conjecture for the fibres of a smooth projective family $f : X \to S$ defined over $\mathcal{O}_{K,N}$. To carry out their strategy, one needs to be able to decide whether the algebraic monodromy group $\mathbf{H}_{Z}$ of any positive-dimensional geometrically irreducible subvariety $Z \subset S_{\mathbb{C}}$ is "large enough", in the sense that a certain orbit of $\mathbf{H}_{Z}$ in a variety of Hodge flags has dimension bounded from below by a certain quantity. In this article we give an effective method for deciding this question. Combined with the effective methods of Lawrence-Venkatesh for understanding semisimplifications of global Galois representations using $p$-adic Hodge theory, this gives a fully effective strategy for solving Shafarevich-type problems for arbitrary families $f$.

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arXiv2021

Sets of Special Subvarieties of Bounded Degree

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb{V} = R^{2k} f_{*} \mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb{V}$ one has the so-called Hodge locus $\textrm{HL}(S) \subset S$, which is a countable union of "special" algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb{V}$ possessing extra Hodge tensors (and so conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline{S}$ and very ample line bundle $\mathcal{L}$ on $\overline{S}$). As a corollary of our algorithm we prove conjectures of Daw-Ren and Daw-Javanpeykar-Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.

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arXiv2021

On the Transcendence of Period Images

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected base $S$, with everything defined over $\overline{\mathbb{Q}}$. Denote by $\mathbb{V} = R^{2i} f_{*} \mathbb{Z}(i)$ the associated integral variation of Hodge structure on the degree $2i$ cohomology. We consider the following question: when can a fibre $\mathbb{V}_{s}$ above an algebraic point $s \in S(\overline{\mathbb{Q}})$ be isomorphic to a transcendental fibre $\mathbb{V}_{s'}$ with $s' \in S(\mathbb{C}) \setminus S(\overline{\mathbb{Q}})$? When $\mathbb{V}$ induces a quasi-finite period map $φ: S \to Γ\backslash D$, conjectures in Hodge theory predict that such isomorphisms cannot exist. We introduce new differential-algebraic techniques to show this is true for all points $s \in S(\overline{\mathbb{Q}})$ outside of an explicit proper closed algebraic subset of $S$. As a corollary we establish the existence of a canonical $\overline{\mathbb{Q}}$-algebraic model for normalizations of period images.

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arXiv2020

Absolute Hodge and $\ell$-adic Monodromy

Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let $σ\in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal{H}_{\mathbb{C}, s}$ above $s \in S(\mathbb{C})$ which lies inside $\mathbb{V}_{s}$, the conjugate vector $v_σ \in \mathcal{H}_{\mathbb{C}, s_σ}$ is Hodge and lies inside $\mathbb{V}_{s_σ}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb{C}}$ containing $s$ whose algebraic monodromy group $\mathbf{H}_Z$ fixes $v$. Using relationships between $\mathbf{H}_Z$ and $\mathbf{H}_{Z_σ}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\mathbf{H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.

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arXiv2020

On the fields of definition of Hodge loci

A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called "an irreducible component of the Hodge locus) for such variations is defined over $\overline{\mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $\overline{\mathbb{Q}}$ to the case of special points.

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arXiv2018

Explicit Rational Group Law on Hyperelliptic Jacobians of any Genus

It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford's classic "Lectures on Theta", where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.

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