Degenerations of maps to projective spaces

TL;DR

This paper explores how degenerations of linear series on smooth projective varieties relate to quiver representations and Grassmannians, revealing their geometric structure and connection to Mustafin varieties.

Contribution

It describes the structure of quiver Grassmannians for simple quivers arising from degenerations, showing they are reduced, rational, and limits of projective spaces.

Findings

Quiver Grassmannians are reduced and local complete intersections.

They are rational and of the same dimension.

These Grassmannians are limits of projective spaces and relate to Mustafin varieties.

Abstract

Degenerations of linear series on smooth projective varieties approaching multicomponent varieties $X$ give rise to certain quiver representations in the category of linear series over $X$, which yield rational maps from $X$ to the corresponding quiver Grassmannians of codimension 1 subspaces. We describe these quiver Grassmannians for the case of the simplest quiver, arising when $X$ has only two components. We prove that they are reduced, local complete intersections whose components are rational of the same dimension. Also, we show that they are limits of projective spaces when they do arise from degenerations, and thus are special fibers of certain Mustafin varieties. Finally, we address a Riemann--Roch question for these quiver representations.