Towards a cluster structure on trigonometric zastava

TL;DR

This paper investigates the Poisson structure of a moduli space related to zastava spaces on a nodal genus 1 curve, comparing it with known structures and proposing a cluster structure conjecture.

Contribution

It introduces a Poisson structure on a moduli space associated with zastava and conjectures a cluster structure arising from generalized minors.

Findings

Computed the Poisson structure in natural coordinates.

Compared the Poisson structure with the trigonometric Poisson structure.

Conjectured a cluster structure on the trigonometric zastava.

Abstract

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.