A construction of Shatz strata in the polystable G2-bundles moduli space using Hecke curves

Let X be a compact Riemann surface of genus g >= 2 and M(G(2)) be the moduli space of polystable principal G(2)-bundles over X. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space M(G(2)) called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of M(G(2)) corresponding to certain family of Harder-Narasimhan types (specifically, those of the form (lambda, mu, 0, -mu, -lambda) with mu < lambda <= 0) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal G(2)-bundles. From this, it was proved that, when the genus of X was g >= 12, these Shatz strata were the disjoint union of a family of G(2)-Hecke curves in M(G(2)) that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of M(G(2)) through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.