Rationality of moduli space of torsion-free sheaves over reducible curve
Let \(k\) be an algebraically closed field of characteristic \(0\) and let \(C\) be a connected nodal curve over \(k\) such that each of its irreducible components has at most \(2\) nodes (the authors call such a curve a \textit{chain-like curve}). Let \(\text{M}(2,\underline{w}, \chi)\) be the moduli space of rank \(2\) torsion-free sheaves \(\underline{w}\)-semistable sheaves of Euler characteristic \(\chi\) over \(C\), where \(\chi\) is an odd integer number and \(\underline{w}\) is a polarization on \(C\) for which semi-stability and stability coincide. The main result of this paper is that the closure of the locus of stable rank \(2\) vector bundles in each irreducible component of \(\text{M}(2,\underline{w}, \chi)\) is rational. Most of the article is devoted to construct a stable family of rank \(2\) vector bundles over \(C\), paramatrized by a non-empty open subset of an affine space, which plays a fundamental role in the proof of the main theorem.
scientific article; zbMATH DE number 6865364
scientific article