Degenerations of Grassmannians via lattice configurations

TL;DR

This paper explores degenerations of Grassmannians through lattice configurations in Bruhat-Tits buildings, connecting quiver representations and Mustafin degenerations, and applies findings to limit linear series smoothing.

Contribution

It introduces a new framework for Grassmannian degenerations using convex lattice configurations, generalizing Mustafin degenerations and extending existing results.

Findings

Explicit description of special fibers as quiver Grassmannians

Equivalence of the construction with Mustafin degenerations for certain configurations

A smoothing criterion for limit linear series on reducible nodal curves

Abstract

We study degenerations of Grassmannians constructed using convex lattice configurations in Bruhat-Tits buildings. Using techniques from quiver representations, we analyze their special fibers, which are explicitly described as quiver Grassmannians. For a class of lattice configurations, called the locally linearly independent configurations, we show that our construction coincide with Mustafin degenerations, thus generalizing a result of Faltings. In such cases, our analysis of special fibers also generalizes results of Cartwright et al. As an application, we prove a smoothing criterion for limit linear series on arbitrary reducible nodal curves.