Timeline for What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Sep 6, 2020 at 14:42 | vote | accept | guest0803 | ||
| Sep 6, 2020 at 6:26 | comment | added | naf | OK, I have posted a simplified version of the argument as an answer. | |
| Sep 6, 2020 at 6:25 | answer | added | naf | timeline score: 2 | |
| Sep 5, 2020 at 17:06 | comment | added | guest0803 | @ulrich I am not sure how to turn your comment into an answer. Would you mind posting it as an answer? | |
| Sep 5, 2020 at 17:05 | comment | added | guest0803 | I think I see your point now. Since you ruled out skyscraper sumands, if $L=IC(\mathbb{L})$ for some local system $\mathbb{L}$ on the punctured curve, we could use the fact that $IC(\mathbb{L} = j_*\mathbb{L}$ on curves, where $j$ is the inclusion of the punctured curve. Since the stalk of $j_*\mathbb{L}$ at the puncture is given by $H^0(\Delta^*, \mathbb{L})$ for small enough punctured discs, and we already knew that the stalks of $L$ are all equidimensional we can conclude that $\mathbb{L}$ has no monodromy around the puncture and hence extends over the puncture as a local system. | |
| Sep 5, 2020 at 14:38 | comment | added | guest0803 | Dear @ulrich, thank you so much for the beautiful explanation. This seems to be an answer and I would happily accept it if you could please post it as an answer instead of a comment. Perhaps I am being slow, but the only point I still do not understand is why $R^1f_*\mathbb{C}$ could not be of the form $IC(\mathbb{L})$ for some non-trivial rank 2 local system $\mathbb{L}$ outside the 4 points. I agree with all the other statements you made. | |
| Sep 3, 2020 at 14:53 | comment | added | naf | cont. In the simplest kind of hyperelliptic surface, there are 4 singular fibres and the local behaviour around each of them is the same. If there are a skyscraper sheaf corresponding to these points, using the Leray spectral sequence to compute $H^1(X,\mathbb{C})$, you get that this is at least 4 dimensional . However, it is easy to see from the description of $X$ as a quotient of a product of elliptic curves that this cohomology group is actually 2 dimensional. (There should, of course, be a purely local proof, but I have not worked that out.) | |
| Sep 3, 2020 at 14:44 | comment | added | naf | OK, here are some details. To make this short, although it is not necessary, I will assume the result in Donu Arapura's answer. It is obvious that $R^0f_*\mathbb{C}$ and $R^2f_*\mathbb{C}$ are constant sheaves, so it suffices to consider $R^1f_*\mathbb{C}$. All the fibres of the map are elliptic curves, but some are not reduced. Using proper base change, the stalks of $L :=R^1f_*\mathbb{C}$ are all isomorphic. To see that it is a local system it suffices to show that $L$ has no summand supported on any point corresponding to a nonreduced fibre. | |
| Sep 3, 2020 at 7:10 | comment | added | guest0803 | Thank you very much for the comment. Unfortunately I am not sure how to see that the decomposition theorem for your example satisfies all the properties in my question. If you could please expand a bit, I will really appreciate it. | |
| Sep 2, 2020 at 10:47 | comment | added | naf | One can find examples where $f$ is not smooth, even with the extra condition in the edit, e.g., this can happen when there are nonreduced fibres. For a specific example, take $X$ to be a hyperelliptic surface and $f$ the natural map to $\mathbb{P}^1$. | |
| S Sep 2, 2020 at 5:03 | history | suggested | CommunityBot | CC BY-SA 4.0 |
Improved the first question
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| Sep 2, 2020 at 4:42 | review | Suggested edits | |||
| S Sep 2, 2020 at 5:03 | |||||
| Sep 1, 2020 at 21:41 | vote | accept | guest0803 | ||
| Sep 6, 2020 at 14:42 | |||||
| Sep 1, 2020 at 20:51 | history | edited | guest0803 | CC BY-SA 4.0 |
added question
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| Sep 1, 2020 at 19:45 | history | became hot network question | |||
| Sep 1, 2020 at 16:54 | answer | added | Donu Arapura | timeline score: 7 | |
| Sep 1, 2020 at 11:51 | review | First posts | |||
| Sep 1, 2020 at 14:16 | |||||
| Sep 1, 2020 at 11:44 | history | asked | guest0803 | CC BY-SA 4.0 |