A desingularization of the moduli space of rank 2 Higgs bundles over a curve (Q2233611)

Let \(X\) be a smooth complex projective curve of genus \(g\geq3\), denote by \(\mathbf{M}_2\) the moduli space of semistable rank 2 Higgs bundles with trivial determinant over \(X\). Let \(\mathbf{M}_2^s\subset\mathbf{M}_2\) be the stable locus, then it's known that \(\mathbf{M}_2\) has singularities along \(\mathbf{M}_2\backslash\mathbf{M}_2^s\). In the present paper, the author constructs a desingularization \(\mathbf{S}\) of \(\mathbf{M}_2\), and shows \(\mathbf{S}\) is a closed subvariety of the moduli space of semistable rank 4, degree 0 parabolic Higgs bundles of certain type \(\mathbf{M}_{4,(a_1,a_2)}^{\mathrm{par}}\), and it is nonsingular, and it contains \(\mathbf{M}_2^s\) as an open dense subvariety. To show the closedness of \(\mathbf{S}\), the author constructs a subset \(\mathbf{S}'\) of \(\mathbf{S}\) and shows \(\mathbf{S}=\overline{\mathbf{S}'}\). To show \(\mathbf{S}\) is nonsingular, the author constructs a subfunctor of the moduli functor of \(\mathbf{M}_{4,(a_1,a_2)}^{\mathrm{par}}\) and shows such functor is formally smooth so that \(\mathbf{M}_2\) represents it. The third argument follows via showing there is a bijection \(\mathbf{M}_2^s\simeq\mathbf{S}'\). Moreover, the author studies Kirwan's desingularization \(\mathbf{K}\) of \(\mathbf{M}_2\), and shows \(\mathbf{S}\) can be obtained via two blow-downs of \(\mathbf{K}\).