SR1

Hilbert schemes classify subschemes of a given projective variety / scheme. They are special cases of Quot schemes which are moduli spaces for quotients of a fixed coherent sheaf. Hilb and Quot are among the first examples of moduli spaces in algebraic geometry, and they are crucial for solving many other moduli problems. I will try to give you a flavour of the subject by sketching the construction of Hilb and Quot and by discussing the role they play in applications, in particular moduli spaces of stable curves and moduli spaces of stable sheaves.

I will discuss some theorems of Chatzidakis, van den Dries, and Macintyre on definable sets over finite fields (Crelle 1992). This includes a geometric decomposition theorem for definable sets and a generalization of the Lang-Weil estimates, and uses model theory of finite and pseudo-finite fields.

If time permits, I shall mention a recent application of this work by Emmanuel Kowalski on new bounds for exponential sums (Israel Journal of Math 2007).

I would also like to mention some connections to the model theory of p-adic and motivic integrals and to general problems on counting and equidistribution of rational points.