Abstract:

If we look at the quadric surface $$x_1x_2 =x_3x_4$$ in complex projective space ${\bf CP}^3$, it turns out to have two one dimensional families of lines on it:$$x_1 = cx_3,\quad x_2 = (1/c) x_4$$and $$x_1 = c x_4, \quad x_2 = (1/c) x_3.$$Actually, it is not hard to show that any smooth surface given by a homogeneous quadratic polynomial has two such families of lines. It is not so obvious that the cubic surface $$x_1^3+x_2^3+x_3^3+x_4^3 =0$$has exactly 27 lines on it--in fact, all smooth cubic surfaces have exactly this number of lines. If we take a surface of degree $d\ge 4$, defined by setting a randomly chosen homogeneous polynomial $F(x_1,x_2,x_3,x_4)$ of degree $d$ equal to 0, then there is a beautiful theorem of Noether and Lefschetz which says that it does not have any lines on it (this is easy) or any other ``interesting" algebraic curves on it (which is the hard part)--this means that all the curves come by intersecting the surface with another surface. Although easy to state, this result was notoriously difficult to prove, and a number of techniques in topology and geometry were developed to find a rigorous proof (As often happens for theorems bearing two names, Noether discovered the result but could not find a rigorous proof, and Lefschetz created some new methods that gave the first proof.)

This is one of the theorems that got me interested in algebraic geometry. It was the first step in what is now a long and really interesting story. It is also one of the first instances in algebraic geometry of variational techniques--using a family to prove a result about ``almost all" cases. It is part of the general area known as Hodge theory and the study of algebraic cycles. This continues to be my favorite part of mathematics, containing some of the deepest and most puzzling open problems in any subject--for example, the celebrated Hodge Conjecture.

Going back to the Noether-Lefschetz theorem, some surfaces of degree $d$ do have interesting curves--for example, $$x_1^d+x_2^d+x_3^d+x_4^d=0$$always has lines on it. After Noether and Lefschetz's work, the conjecture was formulated that the largest family of smooth surfaces which did have interesting curves on them had codimension $d-3$ in the space of all surfaces of degree $d$. The problem is hard because there are an infinite number of possible families of interesting curves that one has to rule out--in fact, the set of surfaces having interesting curves turns out to be dense! Using geometric and analytic techniques, this problem can be reduced to a question in pure algebra about what the minimum rate of growth of an ideal of homogeneous polynomials can be, and then proved using an algebraic theorem of Macaulay and an improvement by Gotzmann. These results and their connection to geometry are what got me interested in commutative algebra.