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

Let Y be a possibly non-connected smooth projective curve, y 1 1 , y 2 1 , ... , y 1 ν , y 2 ν 2 ν different points of Y, r ∈ N , d ∈ Z , δ ∈ Q > 0 , κ ̲ = (κ 1 , ... , κ ν) ∈ Q ≥ 0 ν and e ̲ = (e 1 , ... , e ν) ∈ Z ≥ 0 ν with e i ≤ r . We construct a projective moduli space of (κ ̲ , δ) -(semi)stable singular principal G-bundles of rank r, degree d, with generalized parabolic structure of type e ̲ supported on the divisors D 1 = y 1 1 + y 2 1 , ... , D ν = y 1 ν + y 2 ν . 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 e ̲ on X and the set of isomorphism classes of descending singular principal G-bundles with generalized parabolic structures of type 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 δ -stable singular principal G-bundles of type e ̲ . This generalizes the known results over irreducible curves. [ABSTRACT FROM AUTHOR]