On the Geometry of Hypersurfaces of Low Degrees in the Projective Space

Abstract

The study of the geometry of subvarieties of the complex projective space defined by homogeneous equations of low degrees (and in particular, of hypersurfaces, which are defined by one such equation) is a very classical subject. For example, the fact that a smooth complex cubic surface contains 27 lines was first discovered by Cayley in a 1869 memoir. In another direction, some of these varieties have long been known to be unirational (i.e., parametrizable in a generically finite-to-one fashion by a projective space of the same dimension), but it is only in the 1970s that they were proved to be not rational (i.e., not parametrizable in a generically one-to-one fashion by a projective space of the same dimension). Still today, nobody knows any example of a smooth non rational complex cubic hypersurface of dimension 4. A lot of information has however been gathered on this very rich circle of questions. With very little prerequisites, I will illustrate this very active domain of research by introducing the main tools of the trade and treating in some detail a few examples.