1

I am reading Vakil's FOAG, and in section 5.5, Vakil shows a figure about visualized scheme.enter image description here

My problem is: How can I regard it as a scheme, and which are the associated points and embedded points?

Goldmund
  • 81
  • It looks like the intension is that the two dots and the line inside the plane are the embedded points, while the associated points are the generic points of the same three but also the straight line that is outside, the curved line and the plane. – plop Aug 17 '22 at 14:06
  • If you want actual equations defining it, one approach is to first give equations for (the closures of) each of the six associated points in some ambient affine space (e.g. $\mathrm{Spec} : k[x, y, z]$), and then refer to https://math.stackexchange.com/questions/633256/how-to-define-the-union-of-closed-subschemes-in-an-affine-scheme to take their scheme-theoretic union. – Ravi Fernando Aug 17 '22 at 23:20

1 Answers1

3

I see a scheme with the following properties:

  1. It has three irreducible components, a plane and two curves. One curve intersects the plane (the curvy one), the second curve is non-reduced and disjoint from the other two components (the straight fat curve in the upper left).
  2. The curvy curve component has an embedded point.
  3. The plane component has two embedded points, one of dimension 1 (the fat line) and one of dimension 0.

So in total, the associated points are the generic points of the irreducible components (3 points) and the three embedded points.

red_trumpet
  • 8,515
  • Why the line in the upper left is non-reduced. – Goldmund Aug 18 '22 at 08:03
  • @Goldmund It looks thicker than the curvy curve, which typically indicates non-reducedness (even generically, i.e. the local ring at the generic point is non-reduced). – red_trumpet Aug 18 '22 at 08:26