Timeline for Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?
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| Jun 30, 2023 at 18:17 | comment | added | Dave | Thank you for the comment, @JasonStarr. If $X$ is smooth and $F$ is a divisor, then using the adjunction formula makes sense to me. But even when $X$ is smooth and $F$ not a divisor, I'm not sure how the blowing up process will affect things. Here is what I think so far: if $g:X'\to X$ is the blow up, then we get the exceptional divisor $F'$ and $C'$ with $g_*C'=C$. Now $K_{X'}=g^*K_X+bF'$ for some $b>0$. So $$C\cdot K_X = C'\cdot g^*K_X = C'\cdot K_{X'}-bC'\cdot F' \geq C'\cdot K_{F'}-bC'\cdot F'$$ and $-bC'\cdot F'$ should be negative. I want to say that this is $\geq C\cdot K_F$. | |
| Jun 29, 2023 at 20:31 | comment | added | Jason Starr | Welcome new contributor. Heuristically this seems likely. If $F$ is a divisor, then contractibility of $F$ implies that the intersection number of any irreducible curve against the $\mathbb{Q}$-Cartier divisor class of $F$ is nonpositive; now apply adjunction. If $F$ is not a divisor, consider the blowing up $\widetilde{X}$ of $X$ along $F$ with its exceptional divisor $\widetilde{F}$. Unfortunately, the singularities of $F$ and $X$ intercede. In the toric case, there is always a small resolution of $X$ that is the coarse moduli space of a smooth DM stack (but not so in general). | |
| S Jun 29, 2023 at 17:02 | review | First questions | |||
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| S Jun 29, 2023 at 17:02 | history | asked | Dave | CC BY-SA 4.0 |