Quadrics

Abstract

In this final lecture, we will study in more detail what are perhaps the simplest and most fundamental of all varieties: quadric hypersurfaces. The idea is partly to become familiar with these basic objects and partly to see some of the ideas we have studied in the preceding lectures applied. In the course of this (somewhat lengthy) lecture, we will involve the notions of dimensions, degree, rational maps, smoothness and singularity, tangent spaces and tangent cones, Fano varieties, and families —all in the context of the analysis of one class of objects. This is a much less technically demanding lecture than the last; we are not pushing the boundaries of what we can do with available techniques here, but carrying out a classical and elementary investigation.