# Linear system giving the projective embedding of the tangential variety

I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this topic. Let me briefly sketch the setting: $$X^n \subset \mathbb{P}^N$$ is a projective variety of dimension $$n$$ (we can assume the required properties for $$X$$ ad libitum) and $$TX=\bigcup_{x \in X}\mathbb{T}_{x}X$$ the projective tangential variety, where $$\mathbb{T}_xX$$ is the projective tangent space. Attached to $$X$$ we can also consider its tangent bundle $$\mathcal{T}_X$$. The essential fact is that $$(\mathcal{T}_X)_x$$ is a vector space of dimension $$n$$ while $$\mathbb{T}_xX$$ is a projective space of dimension $$n$$. Of course if we denote with $$\hat{X}$$ the affine cone over $$X$$ then $$(\mathcal{T}_X)_x=T_\hat{x}\hat{X}/\hat{x}$$.

1. First question

With these in mind I want to "construct" a projective bundle $$\mathbb{P}(\mathcal{E})$$ over $$X$$ with some notion of "natural" fiber such that $$\mathbb{P}(\mathcal{E}_x)=\mathbb{T}_xX$$. My first guess would be something of the form $$\mathbb{P}(\mathcal{T}_X \oplus \mathcal{O}_X(i))$$ (for projective dimensional reasons) for some $$i$$ but I'm not sure if this is the right way to think about it.

1. Second question

If such a projective bundle $$\mathbb{P}(\mathcal{E})$$ exists, what is the linear system $$W$$ and line bundle $$\mathcal{L}$$, with $$W \subset H^0(\mathbb{P}(\mathcal{E}),\mathcal{L})$$, that gives me the natural map $$\phi_W:\mathbb{P}(\mathcal{E}) \rightarrow TX \subset \mathbb{P}^N$$?

Thanks in advance for all the responses.

• I don't understand the notation, are the $\mathbb{T}_x X$ allowed to intersect in $\mathbb{P}^N$? Commented Feb 15, 2023 at 20:39
• @BenC yes they are, take for example two tangent spaces to the Veronese surface in $\mathbb{P}^5$.
– gigi
Commented Feb 15, 2023 at 21:13

Consider the Euler exact sequence $$0\rightarrow \mathcal{O}_{\mathbb{P}}\rightarrow \mathcal{O}_{\mathbb{P}}(1)^{N+1}\rightarrow \mathcal{T}_{\mathbb{P}}\rightarrow 0\,.$$Restrict to $$X$$, and pull-back by the inclusion $$\mathcal{T}_X\subset \mathcal{T}_{\mathbb{P}|X}$$. This gives an exact sequence $$0\rightarrow \mathcal{O}_{X}\rightarrow \tilde{\mathcal{T}}_{X}\rightarrow \mathcal{T}_X\rightarrow 0$$with an injective map $$\tilde{\mathcal{T}}_{X}\hookrightarrow \mathcal{O}_{X}(1)^{N+1}$$. The tangential variety is $$\mathbb{P}(\tilde{\mathcal{T}}_{X})$$; it maps to $$\mathbb{P}^N$$ through $$\mathbb{P}(\tilde{\mathcal{T}}_{X})\hookrightarrow \mathbb{P}(\mathcal{O}_{X}(1)^{N+1})=X\times \mathbb{P}^N\rightarrow \mathbb{P}^N$$.
• thank you very much! Do you know if $\hat{\mathcal{T}}_X$ can be characterized in terms of natural bundles attached to $X$ beside being itself an extension of $\mathcal{T}_X$ by $\mathcal{O}_X$?
• I don't know if this is what you are asking for: it is the sheaf of differential operators of degree $\leq 1$ on the line bundle $\mathcal{O}_X(1)$. Note that it definitely depends on $\mathcal{O}_X(1)$, hence on the embedding.