Timeline for Pullback map on global sections surjective
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 18, 2020 at 15:58 | comment | added | user267839 | Thank you very much. A side remark: with same conditions as above for $X,Y$ and $f$ we can moreover say that $f^* \colon H^0(Y,\mathcal{L}) \to H^0(X,f^*\mathcal{L})$ is always injective. what we need I think is that $Y$ is reduced & $f$ dominant | |
| Mar 17, 2020 at 10:43 | comment | added | Sasha | @user7391733: I included an explanation to this question in the answer. | |
| Mar 17, 2020 at 10:42 | history | edited | Sasha | CC BY-SA 4.0 |
added 542 characters in body
|
| Mar 17, 2020 at 10:12 | comment | added | user267839 | Yes, I understand. But why the surjectivity of $H^0(Y,\mathcal{L}) \otimes \mathcal{O}_Y \to \mathcal{L}$ (=regularity, as you said) follows from surjectivity of $f$? I not see it. | |
| Mar 16, 2020 at 15:43 | comment | added | Sasha | Because regularity is equivalent to surjectivity of the morphism $H^0(Y,\mathcal{L}) \otimes \mathcal{O}_Y \to \mathcal{L}$, which, by Nakayama lemma, is enough to check only at closed points. | |
| Mar 16, 2020 at 13:56 | comment | added | user267839 | One question on your argument why rational $Y \to \mathbb{P}(H^0(Y,\mathcal{L})^\vee)$ becomes regular: that's clear pure set theoretically: $f$ is surjective, so we can assign to every $y \in Y$ the image of arbitrary $x \in f^{-1}(y)$ under composition $X \to \mathbb{P}(H^0(X,f^*\mathcal{L})^\vee) \to \mathbb{P}(H^0(Y,\mathcal{L})^\vee)$ simply by "going another way around". Why that (at first glace) pure set theoretical assignment gives rise for regular $Y \to \mathbb{P}(H^0(Y,\mathcal{L})^\vee)$ mapping as morphism or category of schemes? | |
| Mar 16, 2020 at 13:47 | vote | accept | user267839 | ||
| Mar 16, 2020 at 13:20 | history | answered | Sasha | CC BY-SA 4.0 |