Polymatroidal tilings and the Chow class of linked projective spaces

TL;DR

This paper investigates linked projective spaces, showing they have the same Chow class as the diagonal, which advances understanding of their geometric properties and potential degenerations.

Contribution

It proves that linked projective spaces possess the Chow class of the diagonal, providing new insights into their geometric structure and relation to degenerations.

Findings

Linked projective spaces have the Chow class of the diagonal.

They are subschemes of products of projective spaces.

Open question remains on whether they are degenerations of the diagonal.

Abstract

Linked projective spaces are quiver Grassmanians of constant dimension one of certain quiver representations, called linked nets, over special class of quivers, called $\mathbb{Z}^n$-quivers. They were recently introduced as a tool for describing schematic limits of families of divisors. They are subschemes of products of projective spaces of the same dimension. It is an open question whether they are degenerations of the (small) diagonal. We show that they have the Chow class of the diagonal.