examples of interpreting schemes (Eisenbud)

Question

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative Algebra With a viewpoint towards Algebraic Geometry. Assume $k$ is algebraically closed.

1. Apparently $I = (x^2,y) \subset k[x,y]$ defines the geometric object of the origin in $k^2$ along with a (or lots?) of tangent vectors sticking out horizontally. Why?

2. Similarly $I = (x,y)^2 = (x^2,xy,y^2)$ defines the origin along with the ``first order infinitesimal neighbourhood around the origin." Again, why?

I don't understand his reasoning in either case, though apparently he looks at the primary decomposition of the ideals.

Answer 1

This was too long for a comment, but probably isn't the type of answer you are looking for:

This is purely intuition, a nice exposition of which, can be found in Vakil.

The idea is that $(x^2,y)$ has some 'fuzz' (Vakil's terminology) which keeps track of more information than just the point $(0,0)$. It keeps track of information about functions' $x$-component, which, if you think about it, is keeping track of their horizontal tangent vectors. Similarly, $(x,y)^2$ is keeping track of extra information in both the $x$ and $y$-direction. But, intuitively, if your ideal keeps track of two tangent vectors, it keeps track of linear combinations of them. So, since the $x$ and $y$ tangent vectors span the tangent space at $(0,0)$, we see that $(x,y)^2$ retains information for all the tangent vectors, or, in other words, the 'fuzz' (the extra data kept) goes in all directions.

Answer 2

One way to make sense of that is to compute the order of contact of those subschemes of the plane with lines passing through it. The first one intersects once all lines but one. The second one intersects all lines through the point twice.