arXiv subjects

Borys Kadets

Publications and source records attributed to Borys Kadets.

12 recordsLinked to original sources

arXiv2026

Generic Manin-Mumford

Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}^m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$---such as the family of roots of generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, for a finite collection of $\alpha\in \mathbb{Q}$---an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in $\mathbb{A}^2_\mathbb{C}$ has infinitely many points from $\mathcal{S}^2$ if and only if it is of the form $x_1=x_2,$ or $x_1=s$, or $x_2=s$ for a fixed $s \in \mathcal{S}$. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of $\mathcal{S}$ consisting of the roots of unity, and of the Manin--Mumford conjecture. We also show that a similar result holds almost surely for roots of a collection of random polynomials of growing degree and bounded height. The proofs rely on a uniform Galois-theoretic criterion ensuring the unlikely intersection property.

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arXiv2026

Hilbert irreducibility for algebraic points

We study the following problem: given a covering of curves $\phi\colon X \to X_0$ over a number field $k$, and an integer $d$, when is the set \[\{p \in X_0(\overline{k})|\ \mathrm{deg}\ p = d, \text{ and the fiber } \phi^{-1}(p) \text{ is reducible over } k(p)\}\] finite? In case $X$ itself admits infinitely many degree $d$ points, we consider the modified problem where the images of degree $d$ points on $X$ are removed from the set. We prove a number of theorems ensuring a positive answer. As a consequence we show that for a fixed curve $X$ and all sufficiently high-degree indecomposable rational functions $\phi:X \to \mathbb{P}^1$ with $b$ branch points, the set of reducible fibers above degree $d<b/7-2$ points, not containing a degree $d$ point from $X$, is finite.

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arXiv2025

Low degree subvarieties of universal hypersurfaces

We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a family of degree $k$ projective varieties parametrized by $B$. This answers a question raised independently by Farb and Ma. Our main tools consist of a Grassmannian technique due to Riedl and Yang, a theorem of Mumford-Roitman on rational equivalence of zero-cycles, and an analysis of Cayley-Bacharach conditions in the presence of a Galois action. We also show that the large degree assumption is necessary; for $d=3$, rational points are dense in $\text{Sym}^dX_{k(B)}$, and in particular are not collinear.

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arXiv2024

Exponents of Jacobians and relative class groups

We prove a lower bound for the exponent of the relative class group $\mathrm{Pic}^0 X_1/\phi^* \mathrm{Pic}^0 X_2$ for a covering of curves $X_1 \to X_2$ over a finite field $\mathbb{F}_q$. The results improve on the existing best bounds (due to Stichtenoth) in the case $X_2=\mathbb{P}^1$, when the relative class group equals the class group of the function field $\mathbb{F}_q(X_1)$, and are completely new for the genuinely relative situation.

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arXiv2022

Subspace configurations and low degree points on curves

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to $\overline{k},$ admit a map of degree at most $d$ onto $\mathbb{P}^1$ or an elliptic curve. For $d \geqslant 4$ the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of $X$ is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over $k$ for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$.

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arXiv2022

Galois specialization to symmetric points and the inverse Galois problem up to $S_n$

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $π: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in π^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there is a degree $n$, $S_n$-extension $F/\mathbb{Q}$ such that over $F$ the inverse Galois problem for $G$ has a solution.

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arXiv2022

Level structure, arithmetic representations, and noncommutative Siegel linearization

Let $\ell$ be a prime, $k$ a finitely generated field of characteristic different from $\ell$, and $X$ a smooth geometrically connected curve over $k$. Say a semisimple representation of $π_1^{\mathrm{et}}(X_{\bar k})$ is arithmetic if it extends to a finite index subgroup of $π_1^{\mathrm{et}}(X)$. We show that there exists an effective constant $N=N(X,\ell)$ such that any semisimple arithmetic representation of $π_1^{\mathrm{et}}(X_{\bar k})$ into $\mathrm{GL}_n(\bar{\mathbb{Z}_\ell})$, which is trivial mod $\ell^N$, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel's linearization theorem and the $\ell$-adic form of Baker's theorem on linear forms in logarithms.

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arXiv2021

Odoni's conjecture on arboreal Galois representations is false

Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $ϕ_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $ϕ_f$ is surjective. We show that this conjecture is false.

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arXiv2021

Estimates for the number of rational points on simple abelian varieties over finite fields

Let $A$ be a simple abelian variety of dimension $g$ over the field $\mathbb{F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb{F}_q)$. For an arbitrary value of $q$ we prove $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g \leqslant A(\mathbb{F}_q) \leqslant (\lceil(\sqrt{q}+1)^2 \rceil - 1)^{g}$ holds with finitely many exceptions. We compute improved bounds for various small values of $q$. For instance, the Weil bounds for $q=3,4$ give a trivial estimate $A(\mathbb{F}_q) \geqslant 1$; we prove $A(\mathbb{F}_3) \geqslant 1.359^g$ and $A(\mathbb{F}_4) \geqslant 2.275^g$ hold with finitely many exceptions. We use these results to describe all abelian varieties over finite fields that have no new points in some finite field extension.

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arXiv2020

38406501359372282063949 & all that: Monodromy of Fano Problems

A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in $\mathbb{P}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.

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arXiv2020

Sectional monodromy groups of projective curves

Fix a degree $d$ projective curve $X \subset \mathbb{P}^r$ over an algebraically closed field $K$. Let $U \subset (\mathbb{P}^r)^*$ be a dense open subvariety such that every hyperplane $H \in U$ intersects $X$ in $d$ smooth points. Varying $H \in U$ produces the monodromy action $φ: π_1^{\text{ét}}(U) \to S_d$. Let $G_X := \mathrm{im}(φ)$. The permutation group $G_X$ is called the sectional monodromy group of $X$. In characteristic zero $G_X$ is always the full symmetric group, but sectional monodromy groups in characteristic $p$ can be smaller. For a large class of space curves ($r \geqslant 3$) we classify all possibilities for the sectional monodromy group $G$ as well as the curves with $G_X=G$. We apply similar methods to study a particular family of rational curves in $\mathbb{P}^2$, which enables us to answer an old question about Galois groups of generic trinomials.

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arXiv2020

Large arboreal Galois representations

Given a field $K$, a polynomial $f \in K[x]$, and a suitable element $t \in K$, the set of preimages of $t$ under the iterates $f^{\circ n}$ carries a natural structure of a $d$-ary tree. We study conditions under which the absolute Galois group of $K$ acts on the tree by the full group of automorphisms. When $K=\mathbb{Q}$ we exhibit examples of polynomials of every even degree with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of $K=F(t)$ and $f \in F[x]$ in which the corresponding Galois groups are the monodromy groups of the ramified covers $f^{\circ n}: \mathbb{P}^1_F \to \mathbb{P}^1_F$.

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