Quadratic complexes, singular varieties and moduli

Let G be the Grassmannian of lines in P3 embedded in P5 as the Plücker quadric Q. The intersection of Q with a second hypersurface of degree d is what is called a complex of lines of degree d. When we consider the intersection of Q with a second quadratic hypersurface in P5, P, we have a quadratic complex. Let X = Q ∩ P be a quadratic complex that, in this talk, we assume to be non-singular, meaning X is non-singular. The quadric Q contains a 3-dimensional family of planes parametrizing lines in P3, going through a point. These are known in the literature as α-planes. An α-plane, α(p), intersects the quadric P in a conic Kα(p). The singular surface S associated to the quadratic complex X is defined to be the p ∈ P3 such that the plane α(p) corresponding to p intersects the quadric P in a singular conic Kα(p). S = {p ∈ P3 such that rank(Kα(p)) ≤ 2} All this is very classical and can be read for instance in the book by Griffiths & Harris, Principles of Algebraic Geometry. In a joint paper with H. Lange, (D. Avritzer e H. Lange, Moduli spaces of quadratic complexes and their singular surfaces, Geom. Dedicata V. 127 (2007) p. 177-179.), we studied the moduli spaces associated to this objects not only when X is non-singular but also in the singular case. It turns out that there is an equivariant map defined that associates to a quadratic line complex X its singular surface S. The inverse image of a given singular surface S is what is called the Klein variety. In this seminar, I will explain these ideas and their relationship with the moduli space of vector bundles a result that goes back to a famous paper of Narasimhan & Ramanan and was proved independently by P. Newstead.