# Tangent vectors and length two subschemes

can someone explain what the datum of a point $x$ on a scheme $X$ together with a tangent direction has to do with subschemes of length two concentrated in that point? It seems as if there would be some kind of correspondance I don't get.

Thanks a lot!

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See exercise II.2.8 of Hartshorne. –  Andrea Aug 16 '11 at 8:55