Commutative Algebra of n Points in the Plane

Summary

We study questions arising from the geometry of configurations of n points in the affine plane ℂ². We first examine the ideal of the locus where some two of the points coincide, and then study the rings of invariants and coinvariants for the action of the symmetric group Sn permuting the points among themselves. We also discuss the ideal of relations among the slopes of the lines that connect the n points pairwise, which is the subject of beautiful and surprising results by Jeremy Martin.

These lectures address commutative algebra questions arising from the geometry of configurations of n points in the affine plane ℂ². In the first lecture, we study the ideal of the locus where some two of the points coincide. We are led naturally to consider the action of the symmetric group S_n permuting the points among themselves. This provides the topic for the second lecture, in which we study the rings of invariants and coinvariants for this action. As you can see, we have chosen to study questions that involve rather simple and naive geometric considerations. For those who have not encountered this subject before, it may come as a surprise that the theorems which give the answers are quite remarkable, and seem to be hard.

One reason for the subtlety of the theorems is that lurking in the background is the more subtle geometry of the Hilbert scheme of points in the plane. The special properties of this algebraic variety play a role in the proofs of the theorems. The involvement of the Hilbert scheme in the proofs means that at present the theorems apply only to points in the plane, even though we could equally well raise the same questions for points in ℂ^d, and conjecturally we expect them to have similar answers.