# Limit linear series for curves of compact type with three irreducible components

## TL;DR

This paper investigates the properties of limit linear series on curves of compact type with three components, revealing new conditions for unique extensions and providing a constructive method to find all such extensions.

## Contribution

It introduces a characterization for the uniqueness of exact extensions of refined Eisenbud-Harris limit linear series on three-component curves and develops a constructive approach to find all exact extensions.

## Key findings

- Not all refined limit linear series have unique exact extensions.
- A new condition characterizes when a unique extension exists.
- Every refined limit linear series admits at least one exact extension.

## Abstract

Our aim in this work is to study exact Osserman limit linear series on curves of compact type $X$ with three irreducible components. This case is quite different from the case of two irreducible components studied by Osserman. For instance, for curves of compact type with two irreducible components, every refined Eisenbud-Harris limit linear series has a unique exact extension. But, for the case of three irreducible components, this property is no longer true. We find a condition characterizing when a given refined Eisenbud-Harris limit linear series has a unique exact extension. To do this, it is necessary to understand how to construct exact extensions. We find a constructive method, which describes how to construct all exact extensions of refined limit linear series. By our method, we get that every refined limit linear series has at least one exact extension.