Computing S(X) for Ruled Surfaces Defined by Stable Bundles on Curves of Genus 2

This paper discusses the algebra of symmetric tensors for ruled surfaces defined by rank 2 stable vector bundles on curves of genus 2. The algebra is computed for special bundles $E$ when $S^{2}E$ is destabilized by a line bundle and when $S^{3}E$ is destabilized by a bundle of rank 2. A strategy for computation is discussed in regards to the other special bundles $E$. This paper also discusses the generalization of these strategies to higher genus and gives a computation when $E$ is not stable and may be written as a direct sum of line bundles.