Geometric meaning of Selling-reduction data

Determine whether there exist subfamilies of the locus of split Jacobians that restrict the possible sequences of Selling-reduction transformations and, in particular, whether the sequence of transformation parameters has a geometric meaning.

Background

The paper describes the locus of split principally polarized tropical abelian varieties using parameters associated with the elliptic-curve lengths, the torsion order, and the torsion-group isomorphism. To reduce the associated quadratic forms to a chosen cone, Selling's reduction algorithm produces a sequence of integer transformation parameters. The authors explain that, for general split Jacobians, only the existence of such a sequence is known, whereas special subfamilies yield more explicit descriptions because their reduction sequences are constrained.

The unresolved issue is whether additional geometric families can be identified that force particular reduction patterns, such as a fixed sequence length, and whether those patterns encode geometric information rather than being merely algebraic artifacts of the reduction algorithm.

References

If we want to extend our previous approach, the question arises whether there exist other subfamilies of $\mathcal{Q}$ that restrict the possibilities for $(\alpha_1,\beta_1,...,\alpha_n,\beta_n)$ (e.g. that fix $n$). In other words whether $(\alpha_1,\beta_1,...,\alpha_n,\beta_n)$ has geometric meaning.

— Tropical split Jacobians of curves of genus 2 II  (2502.05624 - Cobigo, 8 Feb 2025) in Section 6.2, Moduli space perspective, immediately before the paragraph 'Question (1).'

If we want to extend our previous approach, the question arises whether there exist other subfamilies of $\mathcal{Q}$ that restrict the possibilities for $(\alpha_1,\beta_1,...,\alpha_n,\beta_n)$ (e.g. that fix $n$).

— Tropical split Jacobians of curves of genus 2 II  (2502.05624 - Cobigo, 8 Feb 2025) in Section 6, subsection “A Schottky-type Problem,” paragraph preceding the two Questions (1) and (2)