Stability of symmetric powers of vector bundles of rank two on a curve

Abstract

When a vector bundle $E$ on an algebraic curve is stable, every symmetric power $S^k E$ is known to be stable for sufficiently general $E$. This thesis deals with the question of which $E$ has not stable $S^k E$ in the case where $E$ has rank $2$. As an answer to the question, it is shown that if $S^k E$ is not stable, then $S^m E$ is not stable for some $m=2$, $3$, $4$ or $6$, and moreover, it is shown that if $S^k E$ is not stable, then $S^l E$ is destabilized by a line subbundle for some $l\geq k$. So the stability of $S^k E$ can be rephrased by the existence of a curve with zero self-intersection number on the ruled surface $\mathbb{P}_C(E)$ associated to $E$, and as its corollary, it is shown that every symmetric power $S^k E$ is stable for general $E$. That is, when $E$ has rank $2$, it is possible to remove the 'sufficiently' assumption in the known result. Also, this thesis treats a classification of $E$ in the case of $k=2$ and $3$. When $k=2$, as $E$ with $S^2 E$ being not stable is a vector bundle with orthogonal structure, a relation between known descriptions are investigated, and when $k=3$, it is shown that there exists $E$ with stable $S^2 E$ but not stable $S^3 E$ (when the genus of curve is larger than or equal to $2$). Lastly, this thesis shows that if $S^2 E$ is stable but $S^k E$ is not stable for some $k>2$, then $E$ with trivial determinant is trivialized over an unramified finite covering of given curve. This thesis is based on a paper by the author published in International Journal of Mathematics, and it generalizes the result by removing the assumption on the degree of $E$.