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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!

(p.s. I already read Professor Emerton's answer here Geometric meaning of primary decomposition, but I don't think it addresses my question above.)

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