arXiv subjects

Hyuk Jun Kweon

Publications and source records attributed to Hyuk Jun Kweon.

7 recordsLinked to original sources

arXiv2026

A Uniform Construction of Cohomology Theories of Varieties via Fundamental Groupoids

We prove that the cohomology of a variety is fully recovered from the fundamental groupoids of its Zariski open subsets in various settings. This follows from a new sheaf theory that replaces the target category of Zariski sheaves with a fibered category while retaining the Zariski site as the source. Given a fundamental groupoid functor and a suitable abelian coefficient category, this theory produces the corresponding cohomology theory on varieties, providing a uniform construction of several cohomology theories. The topological fundamental groupoid with abelian groups recovers singular cohomology. The étale fundamental groupoid with discrete abelian groups recovers étale cohomology. The pro-algebraic fundamental groupoid with vector spaces recovers algebraic de Rham cohomology. The pro-algebraic fundamental groupoid with commutative formal groups unifies étale and algebraic de Rham cohomology. In each case, we prove a comparison theorem with the corresponding classical theory or theories. We conjecture that the Nori fundamental groupoid with commutative formal groups unifies étale and p-adic cohomology.

math.AG↗

arXiv2026

Computing Picard Schemes

We present an algorithm to compute the torsion component $\mathrm{Pic}^τX$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^τX$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^τX$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $π^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$.

math.AG↗

arXiv2025

Bornes de torsion et un théorème effectif du pgcd

We prove an effective, probabilistic version of Deligne's `théorème du pgcd' for a smooth, projective, geometrically integral (\textit{nice}) variety $X_{0}\subset \mathbb{P}^{N}$ over $\mathbb{F}_{q}$ of dimension $n$ and degree $D$, obtained via good reduction from a nice variety $\mathcal{X}_{0}$ over a number field $K$ at a prime $\mathfrak{p}\subset \mathcal{O}_{K}$. The main ingredients include bounding torsion in the Betti cohomology of $\mathcal{X}_{0}$, a mod -- $\ell$ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo $\ell$.

math.AG↗

arXiv2023

Maximum Overlap Area of Several Convex Polygons Under Translations

Let $k \geq 2$ be a constant. Given any $k$ convex polygons in the plane with a total of $n$ vertices, we present an $O(n\log^{2k-3}n)$ time algorithm that finds a translation of each of the polygons such that the area of intersection of the $k$ polygons is maximized. Given one such placement, we also give an $O(n)$ time algorithm which computes the set of all translations of the polygons which achieve this maximum.

cs.CG↗

arXiv2023

Maximum overlap area of a convex polyhedron and a convex polygon under translation

Let $P$ be a convex polyhedron and $Q$ be a convex polygon with $n$ vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector $v \in \mathbb{R}^3$ maximizing the overlap area $|P \cap (Q + v)|$ in $O(n \log^2 n)$ time. We then apply our algorithm to solve two related problems. We give an $O(n \log^3 n)$ time algorithm that finds the maximum overlap area of three convex polygons with $n$ vertices in total. We also give an $O(n \log^2 n)$ time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.

cs.CG↗

arXiv2021

Bounds on the Torsion Subgroups of Néron-Severi Group Schemes

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$. Let $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The Néron--Severi group scheme of $X$ is defined by $\mathbf{NS}\, X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$. We give an explicit upper bound on the order of the finite group scheme $(\mathbf{NS}\, X)_{\mathrm{tor}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the finite group $π^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that the torsion subgroup $(\mathrm{NS}\, X)_{\mathrm{tor}}$ of the Néron--Severi group of $X$ is generated by less than or equal to $(\mathrm{deg}\, X -1)(\mathrm{deg}\, X - 2)$ elements in various situations.

math.AG↗

arXiv2019

Bounds on the Torsion Subgroups of Néron-Severi Groups

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. We give explicit upper bounds on the order of the torsion subgroup $(\mathrm{NS} \, X)_{\mathrm{tor}}$ of the Néron-Severi group of $X$. The bounds are derived from an explicit upper bound on the number of irreducible components of either the Hilbert scheme $\mathbf{Hilb}_Q X$ or the scheme $\mathbf{CDiv}_n X $parametrizing the effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on the number of generators of $(\mathrm{NS} \, X)[\ell^\infty]$ uniform as $\ell \neq \mathrm{char}\, k$ varies.

math.AG↗