Timeline for Calculate genus of reducible nodal curve
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| Dec 21, 2023 at 15:19 | comment | added | Wiktor Vacca | The sections of the kernel of $\mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0$ are functions on $C$ vanishing on $C_n\cap D=\Delta$, this gives you a function on $D$ vanishing on $C_n$, hence a section of $\mathcal{O}_D(-\Delta)$, you can check that this defines actually an isomorphism of sheaves. For the second question: it depends on what you know about the curves. What I had in mind was the sequence you wrote. | |
| Dec 17, 2023 at 15:57 | comment | added | user267839 | To put it in a nitshell, I'm wondering if when you say that one can say something about cohom of $\mathcal{O}_D(-\Delta)$ at that stage , if you refering there exactly to the amount of information about it extractible from "natural" les associated to ses $0 \to \mathcal{O}_D(-\Delta) \to \mathcal{O}_D \to \mathcal{O}_{\Delta} \to 0$ only, or do you refer in your comment above to another argument in order to gain information one can gain about cohom of $\mathcal{O}_D(-\Delta)$? | |
| Dec 17, 2023 at 15:45 | comment | added | user267839 | secondly, do I understand your concern correcty, that when you wrote "...then you should be able to say something about those cohomologies." comes from inductive reasoning; namely we can say something about cohom of $\mathcal{O}_D(-\Delta)$ based on assuming we inductively "understood" cohom of $\mathcal{O}_D$ and now analyze the assoc long ex seq containing cohom of $\mathcal{O}_D(-\Delta)$? Or did I misunderstood your suggested strategy? | |
| Dec 17, 2023 at 15:38 | comment | added | user267839 | two points in your explanations are not clear to me. Firstly, why we can identify $\mathcal{I}(C_n)=\mathcal{O}_D(-\Delta)$, assuming we using notations from above where the ideal sheaf $\mathcal{I}(C_n)$ is defined to sit in ses $0 \to \mathcal{I}(C_n) \to \mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0$? | |
| Dec 16, 2023 at 11:54 | comment | added | Wiktor Vacca | In your notation $\mathcal{I}(C_n)=\mathcal{O}_D(-\Delta)$ where $\Delta=C_n\cap D$, then you should be able to say something about those cohomologies. Working in this way you can prove a famous formula by Hironaka which computes the arithmetic genus of a reducible curve, see the answear by Georges Elencwajg in this post math.stackexchange.com/questions/1297433/…. Using this one you can argue by induction on the number of components, supposing as you do $C=C_n\cup D$. | |
| Nov 27, 2023 at 14:33 | history | edited | user267839 | CC BY-SA 4.0 |
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| Nov 27, 2023 at 14:20 | history | edited | user267839 | CC BY-SA 4.0 |
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| Nov 27, 2023 at 14:03 | history | edited | user267839 | CC BY-SA 4.0 |
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| Nov 27, 2023 at 13:58 | history | asked | user267839 | CC BY-SA 4.0 |