Is there an established notation for the linear subvariety tangent to a projective variety $V$ at a point $x$? I've seen this called the "projective tangent space" in some places. The closest thing I've seen to a notation would be something like $$\mathbb{P}(T_{\widetilde{x}} \widetilde{V}),$$ i.e. the projectivization of the (extrinsic) tangent space to the affine cone over $V$ at a point $\widetilde{x}$ corresponding to $x$.
But this is fairly convoluted and still involves overloading some notation and making some implicit identifications.