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 $g\ge 2$. 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 $C$.