Geometry of moduli spaces of rank 2 vector bundles on curves and secant varieties

Let $L$ be a linear bundle on a smooth projective curve and consider the natural rational map from the extension space $\mathbb{P}_L = \mathbb{P}(Ext^1(L,\mathcal{O}))$ to the moduli space $M_{2,L}$ of vector bundles of rank $2$ with determinant $L$. Following Bertram's '92 paper I will construct a new variety $\widetilde{\mathbb{P}_L}$ with a sequence of blow-ups and a morphism to $M_{2,L}$, which allow to calculate dimensions of global sections of linear bundles on $M_{2,L}$ and answer some other questions on geometry of the moduli spaces.