Let \(Y\) be a nodal curve with a single node \(p\) and normalisation a smooth curve \(X\). Let \(p_1, p_2\) be points of \(X\) lying over \(p\). Let \(L\) be a line bundle on \(Y\) and \(L_0\) its pull back to \(X\). A generalised parabolic (\(L_0\)-twisted) Hitchin pair (GPH in short) of rank \(n\) and degree \(d\) is a Hitchin pair \((E,\phi)\) together with a generalised parabolic structure. We recall that \(E\) is a vector bundle of rank \(n\) and degree \(d\) and \(\phi: E \to E \otimes L_0\) is a homomorphism. A generalised parabolic structure consists of a subspace \(F(E) \subset E_{p_1} \oplus E_{p_2}\) of dimension \(n\) with a weight \(\alpha, 0 < \alpha \le 1\). Bhosle had constructed the moduli space \(M_{GPH}\) of \(\alpha\)-semistable GPH and given a proper Hitchin map from it to an affine space \(A\). A GPH is called a good GPH if it preserves \(\phi\), let \(M^{\alpha-st}_{GPH}\) be the moduli space of good \(\alpha\)-stable GPH. There is a morphism \(q: M^{\alpha-st}_{GPH} \to M^{st}_{TFH},\) the latter being the moduli space of stable torsion free (\(L\)-twisted) Hitchin pairs on \(Y\).
The author defines a (stable) Gieseker Hitchin pair data (GHPD in short) generalising the notion of (stable) Gieseker vector bundles by Balaji et al. In case the rank \(n\) and degree \(d\) are coprime, he constructs the moduli space \(M^{st}_{GHPD}\) of these stable objects and shows that it is a normalisation of the moduli space \(M^{\alpha-st}_{GHP}\) of stable Gieseker Hitchin pairs. He shows that the Hitchin map on \(M^{st}_{GHPD}\) is proper. There is a natural morphism \(\pi_0: M^{st}_{GHP} \to M_{TFH}\). The author gives morphisms of moduli schemes \(\pi: M^{st}_{GHPD} \to M^{\alpha-st}_{GHP}\) and \(q_0: M^{st}_{GHPD} \to M^{\alpha-st}_{GPH}\) (for \(\alpha\) close to \(1\) or equal to \(1\)) such that \(\pi_0 \circ \pi = q \circ q_0\).