Generalized parabolic structures over smooth curves with many components and principal bundles over reducible nodal curves

Abstract

Let Y be a possibly non-connected smooth projective curve, $$y_{1}^{1},y_{2}^{1},\ldots ,y_{1}^{\nu },y_{2}^{\nu }$$ $$2\nu $$ different points of Y, $$r\in \mathbb {N}$$ , $$d\in \mathbb {Z}$$ , $$\delta \in \mathbb {Q}_{>0}$$ , $${\underline{\kappa }}=(\kappa _1,\ldots ,\kappa _{\nu })\in \mathbb {Q}^{\nu }_{\ge 0}$$ and $${\underline{e}}=(e_1,\ldots ,e_{\nu })\in \mathbb {Z}_{\ge 0}^{\nu }$$ with $$e_i \le r$$ . We construct a projective moduli space of $$({\underline{\kappa }},\delta )$$ -(semi)stable singular principal G-bundles of rank r, degree d, with generalized parabolic structure of type $${\underline{e}}$$ supported on the divisors $$D_{1}=y_{1}^{1}+y_{2}^{1},\ldots ,D_{\nu }=y_{1}^{\nu }+y_{2}^{\nu }$$ . In case Y is the normalization of a connected and reducible projective nodal curve X, there exists a closed subscheme coarsely representing the subfunctor corresponding to those bundles that descend to X. We prove that the descent operation gives a bijection between the set of isomorphism classes of singular principal G-bundles of type $${\underline{e}}$$ on X and the set of isomorphism classes of descending singular principal G-bundles with generalized parabolic structures of type $${\underline{e}}$$ satisfying certain condition on Y. If the stable locus is dense inside the moduli space of descending singular principal G-bundles, the descent operation induces a birational, surjective and proper morphism onto the schematic closure of the space of $$\delta $$ -stable singular principal G-bundles of type $${\underline{e}}$$ . This generalizes the known results over irreducible curves.