# Connectedness of Brill-Noether loci via degenerations

## TL;DR

This paper investigates the connectedness of spaces of linear series with ramification conditions using degenerations, establishing a criterion that extends previous results to more complex cases.

## Contribution

It introduces new methods in limit linear series theory and provides a criterion for connectedness in spaces with imposed ramification.

## Key findings

- Limit linear series spaces for chains of curves are reduced.
- Spaces with ramification conditions may be disconnected even if positive dimensional.
- A generalized criterion for connectedness is established.

## Abstract

We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.