Searcharxiv⌕ Search

arXiv subjects

Finn Bartsch

Publications and source records attributed to Finn Bartsch.

12 recordsLinked to original sources
arXiv2026

Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell

We construct the first weakly special surfaces that are not Campana-special, including the complement of the plane curve $x^2y^3 = 1$ in $\mathbb{A}^2$. We prove that the set of $\mathcal{O}_{K,S}$-integral points on this surface is non-dense for every number field $K$ and finite set $S$ of finite places of $K$ if and only if Campana's Orbifold Mordell conjecture holds for $(\mathbb{G}_m, \tfrac{1}{2}[1])$. This basic example carries a natural $\mathbb{G}_m$-action, and the quotient stack is an Artin stack parametrizing points on a C-pair. This leads to the introduction of ``Campana stacks'', which encode morphisms of C-pairs in a manner analogous to the role of root stacks for integral points satisfying prescribed divisibility conditions.

math.AG↗
arXiv2026

The map to the orbifold base need not be an orbifold map

We give an explicit example of a fibration $f \colon X \to Y$ between smooth projective varieties whose "orbifold base" $\Delta_f$ in the sense of Campana has the property that the induced morphism $X \to (Y, \Delta_f)$ is not a morphism of C-pairs (i.e., it is not an "orbifold morphism"). We however also show that this cannot happen if $f$ is "neat" and $(Y, \Delta_f)$ is sufficiently well-behaved. Finally, we discuss the implications of this statement towards conjectures of Campana aiming to give algebro-geometric characterizations of those varieties which either admit a dense entire curve or a potentially dense set of integral points.

math.AG↗
arXiv2026

The theorem of Maehara-Severi for maps of general type

We prove a finiteness result for dominant rational maps whose orbifold base is of general type. Our finiteness result generalizes Maehara's theorem that a given variety dominates only finitely many projective varieties of general type up to birational equivalence, and also answers a question of Campana on the finiteness of Bogomolov sheaves. We give several further applications, including finiteness results for maps to curves, abelian varieties, and K3 surfaces.

math.AG↗
arXiv2025

More counterexamples to the Arithmetic Puncturing Problem

We construct examples of threefolds with terminal singularities (resp. surfaces with canonical singularities) which are special in the sense of Campana, have a potentially dense set of integral points, admit a dense entire curve, have vanishing Kobayashi pseudometric, and are geometrically special in the sense of Javanpeykar-Rousseau but whose regular locus fails to have any of these properties. This improves on earlier work by Cadorel-Campana-Rousseau and joint work by the author with Javanpeykar-Levin, where such fourfolds with canonical singularities were constructed, and gives refined answers to questions due to Hassett-Tschinkel and Kamenova-Lehn. Lastly, we show that some of our examples satisfy the weak approximation property and briefly discuss a question on puncturing varieties satisfying strong approximation raised by Wittenberg.

math.AG↗
arXiv2025

On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions

We show that given a simple abelian variety $A$ and a normal variety $V$ defined over a finitely generated field $K$ of characteristic zero, the set of non-constant morphisms $V \to A$ satisfying certain tangency conditions imposed by a Campana orbifold divisor $\Delta$ on $A$ is finite. To do so, we study the geometry of the scheme $\underline{\mathrm{Hom}}^{\mathrm{nc}}(C, (A, \Delta))$ parametrizing such morphisms from a smooth curve $C$ and show that it admits a quasi-finite non-dominant morphism to $A$.

math.AG↗
arXiv2024

Symmetric products and puncturing Campana-special varieties

We give a counterexample to the Arithmetic Puncturing Conjecture and Geometric Puncturing Conjecture of Hassett-Tschinkel using symmetric powers of uniruled surfaces, and propose a corrected conjecture inspired by Campana's conjectures on special varieties. We confirm Campana's conjecture on potential density for symmetric powers of products of curves. As a by-product, we obtain an example of a surface without a potentially dense set of rational points, but for which some symmetric power does have a dense set of rational points, and even satisfies Corvaja-Zannier's version of the Hilbert property.

math.AG↗
arXiv2024

The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture

Starting from an Enriques surface over $\mathbb{Q}(t)$ considered by Lafon, we give the first examples of smooth projective weakly special threefolds which fibre over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere reduced, but non-divisible, fibres and general type orbifold base. We verify that these families of Enriques surfaces (resp. K3 surfaces) are non-isotrivial and compute their fundamental groups by studying the behaviour of local points along certain \'etale covers. The existence of the above threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture. Using these examples, we can also easily disprove several complex-analytic analogues of the Weakly Special Conjecture. Finally, the existence of such threefolds shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005.

math.AG↗
arXiv2024

Parshin's method and the geometric Bombieri-Lang conjecture

In this short survey, we explain Parshin's proof of the geometric Bombieri-Lang conjecture, and show that it can be used to give an alternative proof of Xie-Yuan's recent resolution of the geometric Bombieri-Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety.

math.AG↗
arXiv2023

Weakly-special threefolds and non-density of rational points

We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish fundamental properties of moduli spaces of orbifold maps, and prove a dimension bound for such moduli spaces by using the recent extension of Kobayashi-Ochiai's finiteness theorem for Campana's orbifold pairs.

math.AG↗