Katz

Nets Katz

"Flecnode polynomials and incidence geometry"

We discuss the current state of incidence geometry between points and lines in R^3 and beyond. Much of what is known, even if it can be proved more easily, can be understood as a consequence of Cayley and Salmon's theorem about testing for ruled surfaces. We describe some important but apparently intractable problems that fall below this threshold. We describe the current situation in higher dimensions and mention some questions about varieties having a lot of lines. These are intimately connected with the Kakeya problem.