Non-globally generated bundles on curves

We describe the locus of stable bundles on a smooth genus $g$ curve that fail to be globally generated. For each rank $r$ and degree $d$ with $rg<d<r(2g-1)$, we exhibit a component of the expected dimension. We show moreover that no component has larger dimension and give an explicit description of those families of smaller dimension than expected. For large enough degrees, we show that the locus is irreducible.