In Exercise 6.6.U (a) of Ravi Vakil's FOAG, we are given a scheme $X = \mathbb{C}[x, y]/I$ for some unknown $I$. The one thing we're told about $I$ is the associated points. The exercise then asks us to draw a picture of $X$, including "fuzz".
In Chapter 4.2, we learn to picture nonreduced schemes using "fuzz" in the sense that whenever there is some derivation $D : A \to A/\mathfrak{p}$ which doesn't factor through the corresponding reduced ring $A/\mathfrak{N}$, that the functions $f \in A$ don't just remember the values at $\mathfrak{p} \in X$, but also some "derivatives in directions not in the reduced scheme $\text{Spec}\ A/\mathfrak{N}$", hence we informally picture $\text{Spec}\ A$ to contain some "infinitesimal fuzz" about $\mathfrak{p}$ which is needed to compute these derivatives.
We also learn that fuzz is connected to nilpotents (informally of course), and that nilpotents are connected to embedded primes, and I would assume that the correct solutions to the exercise is to draw fuzz around all the embedded primes. As far as I can tell though, it hasn't been shown anywhere in the book so far that there neccessarilly must be fuzz around embedded primes.
My question is this: Is there some way to formalise these things? I understand that fuzz is not a formal concept, and that it's just meant to guide intuition, but how can we then know where there is fuzz by just knowing the associated primes?