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

Abstract

This paper treats the strict semi-stability of the symmetric powers (SE)-E-k 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 (SE)-E-2 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 (SE)-E-3. Moreover, it is shown that when (SE)-E-2 is stable, every symmetric power (SE)-E-k 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.