Degeneracy loci and unlikely intersections in abelian schemes

Abstract

Let 𝒜→ S be an abelian scheme over a normal quasi-projective variety over a number field, and let 𝒳⊆𝒜 be a positive-dimensional subvariety. For every integer t>0, we prove that, outside the t-degeneracy locus of 𝒳, at most finitely many points lie in flat group subschemes of relative dimension less than t. In particular, this intersection is not Zariski dense whenever 𝒳 is t-nondegenerate. The proof combines a bounded height theorem with uniform large Galois orbit estimates, o-minimal point counting, and mixed Ax--Schanuel, following the Pila--Zannier strategy. The arithmetic input is an explicit bound for the complexity of endomorphism relations, obtained from geometry of numbers and height estimates for abelian varieties. As applications, we formulate a relative Mordell--Lang conjecture, prove it for section images in powers of certain abelian schemes, and establish injectivity of specialization of the group of sections outside a proper closed subset under variation and dimension hypotheses.

ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP
Francesco Campagna, Gabriel A Dill, Robert Wilms
HAL17 Jul 2026PreprintCC BY-NC-ND 4.0Full textGeneral mathematicsHAL
On Thom Polynomials for 𝒜-singularities of parametric curves and degeneracy loci formulas
João Vítor Pissolato
arXiv12 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Motivic classes of Fano schemes of lines on intersections of two quadrics
Lena Ji, Fumiaki Suzuki
arXiv28 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Total Cohomology of Hilbert schemes of Orbisurfaces I: Reduction to Stacky Loci
Shuo Gao
arXiv11 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant
Jesse Chien
arXiv16 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{barroero2026degeneracy,
  title = {{Degeneracy loci and unlikely intersections in abelian schemes}},
  author = {Fabrizio Barroero and Laura Capuano and Tangli Ge and Francesco Tropeano},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06518},
  eprint = {2610.06518},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06518},
}

RIS

TY  - UNPB
TI  - Degeneracy loci and unlikely intersections in abelian schemes
AU  - Fabrizio Barroero
AU  - Laura Capuano
AU  - Tangli Ge
AU  - Francesco Tropeano
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06518
UR  - https://arxiv.org/abs/2610.06518
AB  - Let 𝒜→ S be an abelian scheme over a normal quasi-projective variety over a number field, and let 𝒳⊆𝒜 be a positive-dimensional subvariety. For every integer t>0, we prove that, outside the t-degeneracy locus of 𝒳, at most finitely many points lie in flat group subschemes of relative dimension less than t. In particular, this intersection is not Zariski dense whenever 𝒳 is t-nondegenerate. The proof combines a bounded height theorem with uniform large Galois orbit estimates, o-minimal point counting, and mixed Ax--Schanuel, following the Pila--Zannier strategy. The arithmetic input is an explicit bound for the complexity of endomorphism relations, obtained from geometry of numbers and height estimates for abelian varieties. As applications, we formulate a relative Mordell--Lang conjecture, prove it for section images in powers of certain abelian schemes, and establish injectivity of specialization of the group of sections outside a proper closed subset under variation and dimension hypotheses.
ER  -

Abstract

Related

Cite