Hi everybody,


let $X$ be a smooth projective variety of dimension $n$ and let $f:\mathbb{P}^1 \rightarrow X$ be a morphism.


A theorem of Grothendieck says that the vector bundle $f^{\*}T\_X$ splits as a sum of line bundles, hence we can write $f^{\*}T\_X \cong \bigoplus\_{i=1}^{n}\mathcal{O}\_{\mathbb{P}^1}(a\_i)$ for some integers $a\_i$.


It seems a widely used fact in literature that the number of "negative summands" (i.e. such that $a\_i <0$ in the above decomposition) equals the codimension of the sublocus of $X$ swept out by deformations of $f$.


Does anyone know a precise reference or is able to write down a proof of this fact?


Thank you



Decomposition of $f^{*}T_X$ for a morphism $f:\mathbb{P}^1 \rightarrow X$
The obvious place to look for these kind of issues is János Kollár's [Rational curves on algebraic varieties](http://books.google.com/books?id=oqW3GabJLjgC&printsec=frontcover#v=onepage&q&f=false).


As Jason and JC point out, this is not true as stated. However, there is indeed something resembling this that might be what you are looking for. 
The point is that you cannot expect anything like this for specific curves, but you might for generic ones. There are (at least) two statements that are relevant:


1
-


Let $V\subseteq \mathrm{Hom}(\mathbb P^1,X)$ be an irreducible subvariety. Then the $a\_i$ in your statement are independent of $f$ for a **general** $[f]\in V$. It follows that this way these $a\_i$ are invariants of the family, so one may make the definition $a\_i(V):=a\_i$ for a general $[f]\in V$.  

Reference: [II.3.12.2](http://books.google.com/books?id=oqW3GabJLjgC&lpg=PP1&dq=rational%2520curves%2520on%2520algebraic%2520varieties&pg=PA119#v=onepage&q=rational%2520curves%2520on%2520algebraic%2520varieties&f=false) in [ibid](http://books.google.com/books?id=oqW3GabJLjgC&printsec=frontcover#v=onepage&q&f=false).


2
-


In addition to the above, assume that $a\_i(V)\geq -1$ for all $i$. Then
$$\dim\mathrm{Locus}(V)\geq \#\{i|a\_i(V)\geq 0\}.$$
with equality in characteristic zero. Now, obviously, the negative ones count the codimension.  

Reference: IV.2.7 in [ibid](http://books.google.com/books?id=oqW3GabJLjgC&printsec=frontcover#v=onepage&q&f=false). 


