Configurations of particles with extra information

The Hilbert scheme is a quite general concept in algebraic geometry, describing all subspaces with a specific property. It generalizes the idea of Grassmannians (vector subspaces of given dimension).

Point configurations converging to the origin

The punctual Hilbert scheme of the plane is the simplest example: the configurations of unordered points in the plane $\mathbb{C}^2$. The quotient of $(\mathbb{C}^2)^n$ by the permutation group is singular. The Hilbert scheme is the smallest smooth manifold which desingularizes this quotient (this is called a resolution).

Whenever several points collide, the punctual Hilbert scheme retains as extra information the jet of the curve (i.e. the first terms of the Taylor expansion) along which they collide.

There are several equivalent descriptions of the punctual Hilbert scheme: as space of ideals, resolution or space of commuting matrices.

Equivalent descriptions of punctual Hilbert scheme

The punctual Hilbert scheme has a rich geometric structures, and is used to construct geometric structures.

A video introducing the punctual Hilbert scheme and three different viewpoints:

• Alexander Thomas, Punctual Hilbert Scheme of the Plane, https://www.youtube.com/watch?v=lBLHEn2Dvkw

A research talk with lots of introductury material:

• Alastair Craw, What is the Hilbert Scheme of Points in the Plane?, https://www.youtube.com/watch?v=DfkXEy3b9tw

• Hilbert scheme
• Grassmannians
• Resolution of singularities