How far is the tangent bundle from projective space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:52:19Z http://mathoverflow.net/feeds/question/22390 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22390/how-far-is-the-tangent-bundle-from-projective-space How far is the tangent bundle from projective space? altgr 2010-04-23T20:19:54Z 2010-04-24T04:52:35Z <p>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 <em>projective completion</em> of $T_X\rightarrow X$, then what can be said about the relative dimension of "infinity": $D=\mathbb{P}\backslash T_X$?</p> <p>I apologize if this question is vague. Any thoughts or references will be greatly appreciated.</p> <p>Thanks!</p> http://mathoverflow.net/questions/22390/how-far-is-the-tangent-bundle-from-projective-space/22407#22407 Answer by Sasha for How far is the tangent bundle from projective space? Sasha 2010-04-24T04:52:35Z 2010-04-24T04:52:35Z <p>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 $\omega_X\otimes L$).</p>