Degenerations of the moduli spaces of vector bundles on curves II (generalized Gieseker moduli spaces)

A Degeneration of Moduli of Hitchin Pairs

“…Let T = Spec B be a S-scheme with B a discrete valuation ring and let L be the function field of T which is assumed to be a finite extension of K. Assume that the closed point of T maps to s ∈ S. Let X T = X × S T and let p ∈ X T be the node; let U be a formal neighbourhood of p in X T . We recall ( [15,Page 191] that U is normal with an isolated singularity at p of type A. By the generality of A-type singularities, one can realize U as a cyclic quotient of the affine plane and we can write U = Spec C where C is:…”

“…The main sources for our tools are the papers by Simpson ([24], [25]), and Nagaraj-Seshadri [15] (see also Schmitt [20]).…”

Book Review: Collected papers of C. S. Seshadri. Volume 1. Vector bundles and invariant theory; Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation Theory

“…This provided normal compactifications of the moduli spaces of vector bundles on singular curves. In a couple of interesting papers, degenerations of moduli spaces of vector bundles on curves were studied by Seshadri and Nagaraj [NagSes97], [NagSes99] generalising also results of Gieseker in rank 2. The classification of G-bundles on singular curves turned out to be much more difficult.…”

On the rationality of Nagaraj–Seshadri moduli space

“…It is well known that the moduli space of bundles over curves has a good specialization property, i.e. if a smooth projective curve Y specializes to a projective curve X with nodes as the only singularities, then the moduli space of vector bundles M Y on Y specializes to the moduli space of torsion-free sheaves M X on X [3], [7], [12], [13]. It is known that rationality of projective varieties does not have a good specialization property, for example a family of cubic surface which is rational specializes to a non-rational surface which is birational to E × P 1 where E is a cubic curve .…”