Principal G-bundles on nodal curves

Abstract

In this thesis, we describe the construction of a compact moduli space for principal G-bundles over a node curve X. The process of the construction of these module spaces is based on the work of A. Schmitt. In Chapter 1, we give the background in GIT, coherent sheaves over reduced projective curves, and principal G-bundles. We present some examples for the calculation of the Hilbert-Mumford semistability, which will be important in Chapter 3. We also present a GIT analysis of direct sums of representations that lead to Proposition 1.1.28, which will be crucial in Chapter 3. Chapter 2 is devoted to the construction of SPB (p)-(s)s-P. In section 1, we construct the moduli space of d-semi-stable tensor fields over X, T-(s)s-P (Theorem 2.1.44) following [8,17]. Since our curve X is not irreducible, we must change the rank by the multiplicity in the definition of the d-semi-stability (see subsection 2.1.9). In Section 2, we construct the moduli space of d-semi- stable singular principal G-bundles, SPB(p)-(s)s-P (Theorem 2.2.18). First, we show how to assign a tensor field to each singular principal G-bundle, for what we need to linearize the problem (Theorem 2.2.6). This is done by using a result graded algebras (Lemma 2.2.5). We must show that this assignment is injective (Theorem 2.2.12), using Lemma 1.2.28. In this way, we construct the moduli space as a closed partial scheme of the moduli space of tensor fields. In Section 3, we discuss objects on the normalization of X. In Subsection 1 we construct the moduli space of tensor fields with generalized parabolic structures over a (possibly) non-continuous smooth projective curve Y. The semi-stability condition now depends on v+1 (rational) parameters k1,...,kv, d due to the presence of the additional structure given by the parabolic structure. The moduli space of (k;d)-(semi)stable singular G principal bundles with generalized parabolic structures on Y is constructed as a closed subscheme of the moduli space of tensor fields with a generalized parabolic structure. Finally, we study the stability concepts for large values of the semi-stability parameters. The existence of several minimal points in the curve Y makes it impossible to translate the results of [52]. Here, the technical result which allows us to solve the problem is Proposition 1.1.28. The goal of chapter 4 is to prove the main results, Theorem 4.4.8 and Theorem 4.4.18.