Stability of symmetric powers of vector bundles of rank two with even degree on a curve

TL;DR

This paper investigates the conditions under which symmetric powers of rank two stable vector bundles on a curve are semi-stable or stable, providing classifications and relations between geometric and algebraic properties.

Contribution

It offers a classification of bundles with semi-stable symmetric powers and explores the stability behavior of all symmetric powers based on initial stability.

Findings

$S^2 E$ semi-stable iff $E$ is orthogonal or has a bisection with zero self-intersection

Classification of $E$ with strictly semi-stable $S^3 E$

Stability of $S^k E$ for all but finitely many $E$ in the moduli space

Abstract

This paper treats the strict semi-stability of the symmetric powers $S^k E$ of a stable vector bundle $E$ of rank $2$ with even degree on a smooth projective curve $C$ of genus $g \geq 2$. The strict semi-stability of $S^2 E$ is equivalent to the orthogonality of $E$ or the existence of a bisection on the ruled surface $\mathbb{P}_C(E)$ whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of $E$ with strictly semi-stable $S^3 E$. Moreover, it is shown that when $S^2 E$ is stable, every symmetric power $S^k E$ is stable for all but a finite number of $E$ in the moduli of stable vector bundles of rank $2$ with fixed determinant of even degree on $C$.