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Let $\pi:X\to S$ be a morphism of relative dimension one, where $X$ is a regular surface over a regular scheme $S$ of dimension one (everything over $\mathbb C$). We denote by $\omega_\pi$ the relative dualizing sheaf $\omega_X-\pi^\ast \omega_S$. Let $D$ be any divisor in $X$ and $C\subset X$ a nonsingular curve (Edit:) contained in a fiber. I am interested in understanding why $$\omega_\pi(D)|_C\cong \omega_C(D.C).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ I know that adjunction formula, in this situation, says $\omega_C=\omega_\pi(C)|_C$.

I would start by saying: $$\omega_\pi(D)|_C\cong \omega_\pi|_C(D.C),$$ but I cannot conclude because I do not know whether the latter is isomorphic to $$\omega_\pi(C)|_C(D.C).$$ If this were true, adjunction formula would give the result $(1)$.

Could you please help me to understand why $\omega_\pi|_C(D.C)\cong\omega_\pi(C)|_C(D.C)$?

Brenin
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  • Dear Brenin, what is $\omega_\pi$? Is it the top exterior power of $\Omega_{X/S}$? – Bruno Joyal Oct 25 '13 at 13:29
  • I added the definition I am using. Since everything is regular, I believe it agrees with what you said. – Brenin Oct 25 '13 at 13:37
  • Dear Brenin, I don't understand how your claimed formula can be correct, since the left-hand side depends on $\pi$ but the right-hand side doesn't. Your statement of the adjunction formula is incorrect also: it should say that $\omega_C=\omega_X(C)_{|C}$. Did you mistype, or are you leaving out some hypotheses? –  Oct 25 '13 at 13:38
  • @AsalBeagDubh: Now that you wrote your comment, I realize I should have added that $C$ is contained in a fiber. In that case, I believe my "strong" adjunction formula does hold (as $(\pi^\ast\omega_S)(C)|_C=0$). – Brenin Oct 25 '13 at 13:53
  • Dear Brenin: yes, if C is contained in a fibre, then your formula is correct. Please add that hypothesis to the question. –  Oct 25 '13 at 13:54

1 Answers1

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As written the claimed formula seems to be incorrect. To see this take $D=0$: then the formula would say that $\omega_\pi \vert_C = \omega_C$. But that contradicts what the adjunction formula says in this case, namely $\omega_\pi \vert_C = \omega_C(-C)$.

Suppose $\pi$ has connected fibres. Then the line bundles $\omega_C$ and $\omega_C(-C)$ differ, and hence formula (1) is incorrect, precisely when $C$ is either a component of a reducible fibre of $\pi$ (in which case $C^2<0$) or a multiple fibre of $\pi$ (in which case $O_C(C)$ is torsion but nontrivial in $Pic^0(C)$) – that is, whenever $C$ is not a whole (scheme-theoretic) fibre of $\pi$.

  • So in the case when $C$ is a component of a reducible fiber, does $(1)$ hold or not? – Brenin Oct 25 '13 at 14:30
  • @Brenin: let me edit my answer to make that point clearer. –  Oct 25 '13 at 14:31
  • Now I see, thanks for your clarity. I now believe that, instead, one has $\omega_\pi(D)|_C=\omega_C(-C.C+D.C)$. – Brenin Oct 25 '13 at 15:47
  • @Brenin: yes, that's right, just twisting the adjunction formula by $O_C(D)$. –  Oct 25 '13 at 15:51