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I am not getting the problem how and where to start with, will be pleased for your help: $V$ is a proj.variety in $\mathbb{P}^n$ whose homogeneous radical ideal is generated by homogeneous polynomials $F_1,\dots, F_m$. Show that the projective tangent space to $V$ at $p$ is defined by the homogeneous linear polynomials $dF_1|_p,\dots,dF_m|_p$

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Choose an affine chart around the point $p$, for semplicity $U_0$ and $p=[1:0:\cdots:0]$. So you have to dehomogeneize $F_i$ with respect to the variable $x_0$ and you should compare the projective tangent space and the affine tangent space. –  Andrea May 15 '12 at 17:48
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