Seminários para alunos - Pencils of Quadrics, Hyper-Elliptic Curves, their Jacobians, and Moduli Spaces of Stable Vector Bundles of Rank Two

I will introduce higher-dimensional analogues of pencils of conics (that were introduced by Filipe on April 14) - namely pencils of quadrics. For example, I will explain a classical construction that relates pencils of quadrics with hyper-elliptic curves (double covers of a projective line), and discuss how their linear algebra and geometry is related to various questions about moduli spaces of bundles on these curves, in particular how to construct some forms of Jacobians and moduli spaces of stable rank 2 bundles on a hyper-elliptic curve in terms of a pencil of quadrics and the respective matrix of linear forms (Miles Reid's thesis and Desale-Ramanan's theorem from 1970s).

Just one dimension higher - if we consider pencils of two-dimensional quadrics in a three-dimensional projective space (i.e. 4-times-4 symmetric matrices with entries in homogeneous linear polynomials of two variables) we obtain models of elliptic (genus one) curves, and can also relate it to a version of Poncelet theorem in three-dimensional space. When we consider pencils of 4-dimensional quadrics in 5-dimensional projective space their base locus is an interesting Fano threefold, and the respective spectral curve is a curve of genus two (all of them are hyper-elliptic), with the base locus (Fano threefold) being one of the simplest moduli spaces of vector bundles, and the respective Jacobian of a genus two curve being isomorphic to the variety of lines isotopic with respect to all quadratic forms in the pencil.

This is a somewhat classical geometric topic that easily related to both well-known constructions as well as interesting open problems related to some of my research. So it could be useful to students interested in algebraic geometry or conformal field theory, as well as to those whose research is related to elliptic and genus two curves and their Jacobians, among many others.