# “Projective tangent space” to a projective variety

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.

• I misread "notation" as "notion". Whoops. Anyway, in their books Eisenbud and Harris seem to favor $\mathbb{T}_P(X)$. You can almost make yourself believe that this is a good idea: \mathbb = projective! Except affine space is \mathbb too... – Hoot Jun 18 '15 at 4:10
• Thanks! After some further searching (I was looking for "extrinsic tangent space" at first, and only later realized it's more widely called the "projective tangent space") I've also seen $\widetilde{T}_x V$ in the literature, I guess by analogy with the affine cone thing above. – Daniel McLaury Jun 18 '15 at 12:27