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In order to define the tangent space to a quasi-projective variety $V$ (i.e a locally closed closed subset of $\mathbb{P}^n$ considered with Zariski topology induced from $\mathbb{P}^n$) at a point $p$ we must think of $V$ as an open subset in a closed sub variety $W$ of some fixed projective space. Could any one explain me with a simple example that how the tangent space is at some point of a Quasi Proj.Variety?

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Dear Makuasi, It is not hard to define the tangent space intrinsically. What source are you reading that says otherwise? Regards, –  Matt E May 12 '12 at 16:45
    
I am reading from an invitation to algebraic geometry by karen smith et all..should I write how the defined tangent space at some point of a variety? –  Une Femme Douce May 13 '12 at 12:53

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