Intuition Behind Completion of a Local Ring in AG

I was wondering what the competition of the local ring of a (closed) point on a variety means geometrically. The local ring is just like functions defined locally at the point, but I do not see a way to geometrically interpret the completion of this ring, despite it being used a lot in AG. Can anyone give me a geometric description?

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs

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