On intersections of quadrics

This article is dedicated to Igor Rostislavovich Shafarevich, on his fiftieth birthday by the author, and on his election as an Honorary Member by the London Mathematical Society.
This article reproduces a course of lectures given by the author in the winter semester of 1973 in the Mathematics Faculty of the Moscow State University. The lectures were devoted to a survey of the geometrical results connected with the period mapping of the moduli spaces of various structures. Although this subject has arisen fairly recently, it has already two important problems to its credit: the Lefschetz problem, and the problem of the cubic 3-fold. The theory of the period map is divided up into a local and a global theory. The local theory describes the action of the monodromy group on the periods; it has been the subject of numerous Western articles, since it gives a way of carrying out an induction on the dimension in the proof of the Weil Riemann hypothesis. The global theory is more geometrical, but after the first considerable success (the problem of the cubic 3-fold and of surfaces K3) there have been no more publications devoted to it. The purpose of the present article is to introduce the reader to the global theory by means of a fairly simple new example.