Cohomology of the moduli space of Hecke cycles

Let $X$ be a smooth projective curve of genus $g \ge 3$ and let $M_0$ be the moduli space of semistable bundles over $X$ of rank 2 with trivial determinant. Three different desingularizations of $M_0$ have been constructed by Seshadri \cite{Se1}, Narasimhan-Ramanan \cite{NR}, and Kirwan \cite{k5}. In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in \cite{KL}. As a by-product, we compute the cohomology of the moduli space of Hecke cycles.