turkmath.org

I will start from several classical but very simple, almost high school level, examples of algebraic varieties containing many lines, planes, etc. These varieties are very special, as a typical one from the same family would have no lines at all. This brings up a natural problem of finding the *maximal* possible number of lines, planes, etc. that can be contained in a member of a fixed family (say, hypersurfaces of a given dimension and degree). In general, this problem is wide open, but I will describe an approach that lets one attack it for a wide variety of seemingly unrelated families. Finally, if time permits, I will cite a few recent results.