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

TL;DR

This paper constructs a moduli space for singular principal G-bundles with generalized parabolic structures over reducible curves, extending the theory to nodal curves and analyzing the descent of bundles.

Contribution

It introduces a projective moduli space for singular principal G-bundles with generalized parabolic structures over reducible curves, including the case of nodal curve normalization.

Findings

Constructed a projective moduli space for these bundles.

Proved the descent operation induces a birational, surjective, and proper morphism.

Extended the theory to reducible nodal curves and their normalization.

Abstract

Let $Y_{1},\dots,Y_{l}$ be smooth irreducible projective curves and let $Y$ be its disjoint union. Given a semisimple reductive algebraic group $G$ and a faithful representation $\rho:G\hookrightarrow \textrm{SL}(V)$ we construct a projective moduli space of $(\underline{\kappa},\delta)$-(semi)stable singular principal $G$-bundles with generalized parabolic structure of type $\underline{e}$. In case $Y$ is the normalization of a connected and reducible projective nodal curve $X$, there is a closed subscheme coarsely representing the subfunctor corresponding to descending bundles. We prove that the descent operation induces a birational, surjective and proper morphism onto the schematic closure of the space of $\delta$-stable singular principal $G$-bundles whose associated torsion free sheaf is of local type $\underline{e}$.