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.


2. **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.



Linear system giving the projective embedding of the tangential variety
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 $\mathbb{P}(\tilde{\mathcal{T}}\_{X})\hookrightarrow \mathbb{P}(\mathcal{O}\_{X}(1)^{N+1})=X\times \mathbb{P}^N\rightarrow \mathbb{P}^N$.


