Abstract:

A quadric surface in P^3 can be covered completely by lines. A cubic surface famously contains 27 lines. A quartic surface is an example of a K3 surface, which, at first glance, appears as though it may contain no lines, or in fact no rational curves of any degree. Surprisingly, algebraic K3 surfaces contain lots of curves, and moreover, if you package the finite numbers into a generating function by their degree, you get a quasi-modular form. In this talk I’ll try to sketch what is going on here.