Degenerations of the moduli spaces of vector bundles on curves. II. (Generalized Gieseker moduli spaces) (Q1306818)

For part I see \textit{D. S. Nagaraj} and \textit{C. S. Seshadri}, ibid. 107, No. 2, 101-137 (1997; Zbl 0922.14023). Let \(X_0\) be an irreducible projective curve of arithmetic genus \(g\geq 2\) whose singularity is one ordinary point. The authors give a generalisation of Gieseker's construction for arbitrary rank. They construct a birational model \(G(n,d)\) of the moduli space \(U(n,d)\) of stable torsion free sheaves in the case \((n,d)=1,\) such that \(G(n,d)\) has normal crossing singularities and behaves well under specialization, i.e. if a smooth projective curve specializes to \(X_0,\) then the moduli space of stable vector bundles of rank \(n\) and degree \(d\) on \(X\) specializes to \(G(n,d)\). This generalizes an earlier work of Gieseker in the rank two case. Theorem 1. There exists a canonical structure of a quasi-projective variety on \(G(n,d)\) and a canonical proper birational morphism \(\pi _{*}:G(n,d)\rightarrow U(n,d)_s\) onto the moduli space of stable torsion free sheaves on \(X_0.\) The singularities of \(G(n,d)\) are (analytic) normal crossings. If \((n,d)=1\), \(G(n,d)\) is projective, since \(U(n,d)_s=U(n,d)\) is projective.