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Let $X$ be a smooth projective curve of genus $g \geq 2$ and $M$ be themoduli space of rank 2 stable vector bundles on $X$ whose determinants areisomorphic to a fixed odd degree line bundle $L$. There has been a lot of worksstudying the moduli and recently the bounded derived category of coherentsheaves on $M$ draws lots of attentions. It was proved that the derivedcategory of $X$ can be embedded into the derived category of $M$ by the secondnamed author and Fonarev-Kuznetsov. In this paper we prove that the derivedcategory of the second symmetric product of $X$ can be embedded into derivedcategory of $M$ when $X$ is non-hyperelliptic and $g \geq 16$.

Symmetric products and moduli spaces of vector bundles of curves