Is there a general theory of embeddings of the (total variety of) the tangent bundle on a (nonsingular) projective variety into projective space? I suppose what I really mean is (and to be more precise about the projective space), if $\mathbb{P}$ is the *projective completion* of $T\_X\rightarrow X$, then what can be said about the relative dimension of "infinity": $D=\mathbb{P}\backslash T\_X$?


I apologize if this question is vague. Any thoughts or references will be greatly appreciated.


Thanks!



How far is the tangent bundle from projective space?
The simplest way to get a "projective completion" is to consider the projectivization on $X$ of $T\_X \oplus L$ for some line bundle $L$ on $X$. In this case the complement will be the projectivization of $T\_X$ and will have codimension 1. Sometimes you can contract this completion to get smaller complement, e.g. if $X$ is a curve (choose $L$ in such a way that $\omega\_X\otimes L$ is very ample, then $P(T\_X \oplus L) = P(T\_X\otimes L^{-1} \oplus O)$ which is a blowup of the projective cone over $X$ in the embedding given by $\omega\_X\otimes L$).


