A desingularization of the moduli space of rank 2 Higgs bundles over a\n curve

Let $X$ be a smooth complex projective curve of genus $g\\geq 3$. Let\n$\\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with\ntrivial determinant over $X$. We construct a desingularization $\\mathbf{S}$ of\n$\\mathbf{M}_2$ as a closed subvariety of a moduli space. We prove that\n$\\mathbf{S}$ is a nonsingular variety containing the stable locus of\n$\\mathbf{M}_2$ as an open dense subvariety. On the other hand, there is another\ndesingularization $\\mathbf{K}$ of $\\mathbf{M}_2$ obtained from Kirwan's\nalgorithm. We show that $\\mathbf{S}$ can be obtained after two blow-downs of\n$\\mathbf{K}$.\n