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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

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    $\begingroup$ Offtopic: The classification of vector bundles on the projective line was only rediscovered by Grothendieck. Dedekind-Weber already found it in 1892. $\endgroup$
    – Martin Brandenburg
    Commented Jan 28, 2012 at 10:15
  • $\begingroup$ @Luca Benzo. This is interesting. Could you explain in more detail what you mean by "the sublocus of $X$ swept out by deformations of $f$" ? $\endgroup$
    – Damian Rössler
    Commented Jan 28, 2012 at 10:20
  • $\begingroup$ Hi Damian, set-theoretically this is just the set of points $x \in X$ such that there exists a deformation of the curve $C \doteq f(\mathbb{P}^1)$ passing through $x$. I used the word "locus" cause I didn't try to prove that it is a (closed) subvariety in $X$, although this should be true (there a similar exercise in Debarre's book Higher dimensional algebraic geometry). $\endgroup$
    – Luca Benzo
    Commented Jan 28, 2012 at 10:41
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    $\begingroup$ What you write is simply untrue. I suggest you look for "lines of type II" on cubic threefolds, e.g., as discussed in the paper of Clemens and Griffiths. Where in the literature does this "widely used fact" appear? $\endgroup$
    – Jason Starr
    Commented Jan 28, 2012 at 14:42

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The obvious place to look for these kind of issues is János Kollár's Rational curves on algebraic varieties.

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 in ibid.

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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.

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  • $\begingroup$ Thank you! Actually I knew fact 1 you cite, I simple didn't think over it while writing the question since in my context I always work up to a general deformation of f. Moreover $∑a_i$ is independent of $f$ for every $[f]∈V$, which is consistent with example suggested by Jason Starr. I didn't know the second fact you cite, now I check it $\endgroup$
    – Luca Benzo
    Commented Jan 30, 2012 at 10:13

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