# Geometric explanation of primary ideals

I would like to know how to think about primary ideal geometrically. Vaguely speaking, I think it's an irreducible closed subscheme with some "infinitesimal" data - however, I am not sure how to make this precise.

In particular, when I try to play with this example $\mathbb{C}[x,y,z]/(xy-z^2)$ (from Atiyah-Macdonald), it is easy to check that $(x,z)^2$ is not primary even though it is really the ruling $(x,z)$ together with some "infinitesimal" data. I am guessing the geometric obstruction lies in the non-smoothness at the origin, which makes $(x,z)^2$ not a "real" infinitesimal neighborhood around the origin, but again I don't know how to make this precise.

Another example is $(x^2,xy) \subset \mathbb{C}[x,y]$. Is there a conceptual explanation that this is not a primary ideal by drawing pictures?

Any insight is welcome. Thanks!

For a primary ideal $$I$$, if we take the radical $$\sqrt I$$, we get a prime ideal $$\mathfrak p$$ which corresponds to the underlying irreducible component $$Z=V(I)$$. Moreover we have some "infinitesimal"/non-reduced data along $$Z$$. Being primary says that this non-reduced data must be "uniformly" supported over all of $$V(I)$$ and not on some proper subvarieties.

Precisely this means if we have a non-reduced element $$a$$ (or any element for that matter), then for any subvariety $$W\subsetneq Z$$ with corresponding prime ideal $$\mathfrak q\supsetneq\mathfrak p$$, $$a$$ cannot be killed by $$\mathfrak q$$ (since $$\mathfrak q$$ are the functions vanishing on $$W$$, $$\mathfrak q$$ killing $$a$$ would mean that $$a$$ is supported on $$W$$). This gives the equivalent definition for $$I$$ being primary (when $$A$$ is noetherian): $$A/I$$ having only one associated prime.

In the example of $$I=(x^2,xy)\subset k[x,y]$$, geometrically we are looking at the intersection of the "double $$y$$-axis" $$(x^2)$$ and a "cross" $$(xy)$$: this gives the $$y$$-axis with some non-reduced structure in the $$x$$-direction at the origin. As $$\sqrt I=(x)$$, the underlying variety $$Z$$ is the $$y$$-axis. The function $$x$$ is zero along the $$y$$-axis, except at the origin $$W$$, where it need to vanish with order 2. But then it can be killed by some function defining $$W$$ in $$Z$$, say $$y$$. So we have $$x\notin I$$, $$y\notin \sqrt{I}$$, but $$xy\in I$$.

Similarly, in the example $$I=(x,z)^2\subset k[x,y,z]/(xy-z^2)$$, we intersect the cone $$xy=z^2$$ with the "double $$y$$-axis": we get "uniform" reduced structure in the $$z$$-direction along the whole $$y$$-axis, but with extra reduced structure in the $$x$$-direction at the origin. So again we can take the function $$x$$, which is supported at the origin and killed by $$y$$.