We give an algebraic approach to the study of Hitchin pairs and prove the tensor product theorem for Higgs semistable Hitchin pairs over smooth projective curves defined over algebraically closed fields of characteristic zero and characteristic , with satisfying some natural bounds. We also prove the corresponding theorem for polystable Hitchin pairs.
We study the behavior of the Horrocks-Mumford bundle FHM when restricted to a plane P2 ⊂ P4, looking for all possible minimal free resolutions for the restricted bundle. To each of the 6 resolutions (4 stable and 2 unstable) we find, we then associate a subvariety of the Grassmannian G(2,4) of planes in P4. We thus obtain a filtration of the Grassmannian, which we describe in the second part of this work.
We review a construction of Ellingsrud-Strømme relating instantons of charge n on the ordinary projective space and theta-characteristics on a plane curve of degree n with some extra-structure.