In this talk, I describe how to apply algebraic geometric techniques to solve specific incidence problems. First, we focus on an incidence problem of Bourgain. It asserts that given a set of N^2 lines and a set of points S in a real 3-space such that each line contains at least N points of S, then the size of S is O(N^3) as expected. This statement was recently proven by Guth and Katz. I show how to generalize this theorem to 3- spaces over any arbitrary field and I explain some of the special features of algebraic geometry in characteristic p.

Second, I describe another incidence problem, the joints problem with multiplicities. Given a set of lines L in 3-space we say that a point is a joint if there are 3 lines of L through the point so that their directions span the whole space. I show how to apply estimates on the arithmetic genus of local complete intersection curves to give an upper bound for the number of joints with multiplicities.

The talk is based on work joint with Jordan Ellenberg.