Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.02569

Tropical geometry in torus bundles

Emily Dodwell
View original source ↗Read source document ↗Export metadata ↓

Abstract

We formulate and prove an analogue of the balancing condition for the tropicalization of a curve in a torus bundle $X$. We find that the usual balancing condition fails when the bundle is non-trivial and that the failure is captured by the first Chern classes of the line bundles associated to $X$. We discuss a geometric perspective of tropicalization where the weights arise naturally from intersection theory. The relations between divisors on a toric variety bundle put constraints on these weights which leads to the balancing condition.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emily Dodwell. 2025-08-19. Tropical geometry in torus bundles. https://arxiv.org/abs/2509.02569

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.

math.AG↗
arXiv2026

A refinement of the coherence conjecture of Pappas and Rapoport

The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. We refine this equality of dimensions to an isomorphism of representations. The comparison is established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line, ramified at 0. We further strengthen this comparison by equipping any line bundle on the global Schubert variety of $\mathcal{G}$ with a unique equivariant structure under the global jet group scheme. As an application, we obtain new relations among affine Demazure modules.

math.AG↗
arXiv2026

Refined tropical invariants and characteristic numbers

We prove that the Göttsche-Schroeter and Schroeter-Shustin refined invariants specialize at $q=1$ to the enumeration of rational, resp. elliptic complex curves on arbitrary toric surfaces matching constraints that consist of points and of points with a contact element. Furthermore, we show that the refined invariant extends to the case of any genus $g\ge2$ and either one contact constraint or points in Mikhalkin position, and it again specializes to the corresponding characteristic number at $q=1$. In the appendix we show the limitations of extending this count to a more general setting.

math.AG↗