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Assume we have a birational morphism $\pi:X\to Y$ between smooth projective varieties with non-empty exceptional locus $E$ and irreducible components $E_1,\dots,E_r$.

My question:

Is it possible that there are some numerical relations between these components seen as divisors? For example, is it possible that $E_i\equiv E_j$ for $i\ne j$?

In the case $\pi$ is a blow-up morphism, the different exceptional divisors are never numerically equivalent to each other. Indeed, a line contained in one of them has negative intersection with it but no intersection with all the others.

The answer would be negative if all the birational morphisms could always be written as a composition of blow-up morphisms. I found in Hartshorne, Remark V.5.4.4, that this is false in dimension $\ge3$. However, I checked the references, but I was not able to find a counterexample there! Could anyone enlighten me on that?

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  • $\begingroup$ Welcome new contributor. By the general theory of Picard groups, every numerically trivial divisor on $X$ is, after multiplication by sufficiently positive and divisible integers, linearly equivalent to the pullback from $Y$ of a divisor class that is algebraically equivalent to zero. If the divisor on $X$ is supported on the exceptional locus, then the divisor on $Y$ is even linearly equivalent to zero. $\endgroup$ Commented Jul 8 at 17:52
  • $\begingroup$ Since a rational function on a normal projective variety is uniquely determined (up to invertible scalar) by its corresponding principal divisor, the principal divisor on $Y$ of a principal divisor on $X$ supported in the exceptional locus is the divisor of an invertible scalar. Hence the principal divisor on $X$ is also the divisor of an invertible scalar. $\endgroup$ Commented Jul 8 at 18:01

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This is not possible, even under weaker hypotheses. One argument uses the "negativity lemma", Lemma 3.39 of Kollár--Mori. Part (1) of that lemma says:

Let $h \colon Z \rightarrow Y$ be a proper birational morphism between normal varieties. Let $-B$ be a $h$-nef $\mathbb Q$-Cartier $\mathbb Q$-divisor on $Z$. Then $B$ is effective iff $h_*(B)$ is.

Now suppose there is a nontrivial numerical relation $\sum_i \alpha_i E_i=_{num}0$ between your exceptional divisors. Let $B = -\sum_i \alpha_i E_i$. Then $-B=\sum_i \alpha_i E_i$ is nef, and $\pi_*(B)=0$ is effective, so the lemma says that $B$ is effective. But now since $B$ itself is nef too, we can swap the roles of $B$ and $-B$ to conclude that $-B$ is effective also. This is a contradiction unless $B=0$.

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    $\begingroup$ Thank you so much @Lazzaro Campeotti. I checked the reference and your proof is perfect. However, the Lemma is rather difficult to prove and I suspect that the problem can be solved by a lighter argument. Maybe $h_*B=0$ plus $B\equiv 0\implies B=0$ has an easy proof? $\endgroup$ Commented Jul 8 at 15:50
  • $\begingroup$ Dear @ensdromielo: I would say the proof of the negativity lemma is somehow just an elaboration of the Hodge Index Theorem. In the case $h_*B=0$, i.e. $B$ is supported on the exceptional locus, essentially all we do is apply Hodge Index Theorem to a suitable 2-dimensional subvariety of the base. Anyway, Jason Starr has provided a more elementary answer. $\endgroup$ Commented Jul 9 at 13:40
  • $\begingroup$ It appears to me that the proof in Koll'ar -- Mori does not require characteristic zero (although they are a little unclear about their hypotheses). They do use resolution of surface singularities, but that holds in all characteristics. $\endgroup$ Commented Jul 12 at 14:52
  • $\begingroup$ @JasonStarr: Dear Jason, you are right that I was sloppy about checking that the hypotheses of my answer matched those of the OP's question. I think you are also right that the proof goes through in any characteristic. If one wants a proof which is explicitly stated to be valid in any characteristic, one can be found in Section 2.3 of Birkar, "Existence of flips and minimal models for 3-folds in char p", available at arxiv.org/pdf/1311.3098. $\endgroup$ Commented Jul 14 at 15:45

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