Vector bundles on trees of smooth rational curves

TL;DR

This paper characterizes when vector bundles on a tree of rational curves can be specialized from bundles on the projective line, extending previous rank 2 results to higher ranks.

Contribution

It provides necessary and sufficient conditions for the specialization of vector bundles from 1 to trees of rational curves, generalizing prior work on rank 2 bundles.

Findings

Established criteria for bundle specialization on trees of rational curves.

Extended Coskun's rank 2 specialization results to higher ranks.

Provides a comprehensive framework for understanding vector bundle degenerations.

Abstract

Given a vector bundle $E$ on a tree of smooth rational curves $C$, we give necessary and sufficient conditions for a vector bundle $E'$ on $\mathbb{P}^1$ to specialize to $E$ on $C$, generalizing the rank 2 case, due to Coskun.