Singular intersections in split semiabelian varieties

Abstract

Let 𝒢 be a split semiabelian variety of dimension n ≥ 2 and let 𝒞 ⊂ 𝒢 an irreducible curve, where everything is defined over ℚ̅. We prove that, under the natural hypothesis that 𝒞 is not contained in any proper algebraic subgroup of 𝒢, the set of points P of 𝒞 lying in an algebraic subgroup H which intersects the curve tangentially at P is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.

Uniform Mordell--Lang conjecture for semiabelian varieties
Zhaobo Han, Wenbin Luo, Jiawei Yu
arXiv3 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Equidistribution for semiabelian varieties over number fields
Wei Xue
arXiv30 Aug 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties
Khai-Hoan Nguyen-Dang
arXiv31 Aug 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Smoothing of singular intersections of ellipsoids: pyramitoid
Enrique Artal Bartolo, Santiago López de Medrano, María Teresa Lozano Imízcoz
arXiv11 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
G₀ of complete split toric varieties
Zeyu Shen
arXiv23 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{ballini2026singular,
  title = {{Singular intersections in split semiabelian varieties}},
  author = {Francesco Ballini and Laura Capuano and Nicola Ottolini},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35307},
  eprint = {2609.35307},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35307},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Singular intersections in split semiabelian varieties
AU  - Francesco Ballini
AU  - Laura Capuano
AU  - Nicola Ottolini
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35307
UR  - https://arxiv.org/abs/2609.35307
AB  - Let 𝒢 be a split semiabelian variety of dimension n ≥ 2 and let 𝒞 ⊂ 𝒢 an irreducible curve, where everything is defined over ℚ̅. We prove that, under the natural hypothesis that 𝒞 is not contained in any proper algebraic subgroup of 𝒢, the set of points P of 𝒞 lying in an algebraic subgroup H which intersects the curve tangentially at P is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.
ER  -

Abstract

Related

Cite