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 ≥ 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 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. [ABSTRACT FROM AUTHOR]