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

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 \(C\).