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When I first began on this site MathOverflow took questions at all levels. Then Math.SE was created and the level of MO has crept up steadily with Math.SE not necessarily absorbing the remainder questions. In my opinion, there seems to be currently a gap between upper level of questions answered on Math.SE and those asked on MO.

In any case, what is the current stance on early graduate level questions. E.g. "What is the pullback of the line bundle $\mathcal{O}_{\mathbb{P}^4}(n)$ to the surface $M$?" Obviously not research level but not elementary either. Where does it go? MO, Math.SE

I also found this from 2 years ago Answering not research level questions answer seems pretty firm that MO is only for research questions, but I am asking for second opinion.

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    $\begingroup$ Is the e.g. a real example? can you link to some old MO questions you refer to? In any case, My view is that early graduate level questions are ok for MO. $\endgroup$ – Gil Kalai Aug 27 '15 at 17:21
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    $\begingroup$ The other meta-question you link to is quite different. That question asks about answering questions deemed as off-topic by the person that might answer. You by contrast ask what is on/off-topic. The word "research level" is a bit of a red herring. $\endgroup$ – user9072 Aug 27 '15 at 17:48
  • $\begingroup$ @GilKalai I added quite a bit more detailed and asked successfully on MO - but not answers quite yet. mathoverflow.net/questions/215775/… $\endgroup$ – john mangual Aug 27 '15 at 17:55
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    $\begingroup$ Good luck! The question seems certainly appropriate. I was wondering about the question "What is the pullback of the line bundle $\mathcal{O}_{\mathbb{P}^4}(n)$ to the surface $M$?" I could not find it. $\endgroup$ – Gil Kalai Aug 27 '15 at 18:03
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    $\begingroup$ Let me just say that one should not underestimate MSE. In algebraic geometry at least (relevant to these examples) there are many strong contributors who would certainly handle both the hypothetical example question above and the actual question you posted at mathoverflow.net/questions/215775/… If you want to avoid "policing" then it might work better just to ask at MSE first. $\endgroup$ – potentially dense Aug 29 '15 at 7:41
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    $\begingroup$ Some questions do well on MO and others don't, and probably there are reasons for this, but "research level" or lack of it is just one aspect among many. Quite a few very popular questions are well below research level in any possible interpretation of the term. $\endgroup$ – Christian Remling Aug 29 '15 at 16:20
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    $\begingroup$ Graduate level is under-defined. For example my university re-coded 1/4 the math units I took as an undergraduate as part of graduate courses, making there textbook arguably graduate-level books. It varies by country and by university. $\endgroup$ – Lyndon White Jun 2 '16 at 2:26
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The official help page at https://mathoverflow.net/help/on-topic says

MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books.

To me, that seems to say clearly that a beginning graduate student reading a GTM can ask for help about points that confuse them, provided that they make serious effort to think about it first (and also make a serious effort to digest any answers, before asking follow-up questions).

Although it seems to me that the help page is clear, one could ask separately whether we are happy with what it says. Personally I am happy, but others may not be.

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    $\begingroup$ I would say it is clear, and yet so many people get it wrong. The "police" on MathOverflow put so many questions on hold these days. Sometimes I am not sure why. $\endgroup$ – john mangual Aug 27 '15 at 17:57
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    $\begingroup$ "can ask for help about points that confuse them, provided that they make serious effort to think about" I agree, but want to stress that it is important not only that the asker made the effort but also that they document this effort in some way when phrasing the question. $\endgroup$ – user9072 Aug 27 '15 at 18:13
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    $\begingroup$ I've always felt that the dividing line should be at the level of a strong graduate student entering the second year (and having the sense to know when a question is not one they can easily answer themselves). With this line, some GTMs would be totally off limits, while others would be totally acceptable. $\endgroup$ – Steve Huntsman Aug 27 '15 at 19:11
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    $\begingroup$ Neil, @Steve, what does GTM stand for? $\endgroup$ – Asaf Karagila Aug 27 '15 at 21:13
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    $\begingroup$ @Asaf, Graduate Text in Mathematics (I think). $\endgroup$ – Gerry Myerson Aug 27 '15 at 23:10
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    $\begingroup$ @Gerry, Thanks Mate. Also GTM. $\endgroup$ – Asaf Karagila Aug 28 '15 at 0:04
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It is almost four years that I wrote the following in a thread on the relation of math.SE and MO:

[...] it is from an MO persepective my opinion that a certain type of question that in my opinion used to be more present on MO, and I assume this is precisely the type that now might be asked on MSE, starts to be lacking on MO. What I mean are perhaps too vaguely described "solid graduate level questions" (not the most standard textbook-excercise question from some graduate Analysis or Algebra course but, quoting the FAQs emphasize mine, "the sorts of questions you come across when you're writing or reading articles or graduate level books"; in other words question that are a bit beyond what one can find in any book on the subject but questions that might come up when thinking beyond what one read).

I believe that, ultimately, it is the smaller number of this type of questions that causes the at first glance paradoxical situation that on the one hand some people say that the level of MO significantly increased while at the same time I see some (at the strict/high-level end) users say that MO used to be better.

As Neil Strickland remarked questions arising in the study of graduate level books are on-topic. Arguably, it also should not be forgotten that the site was founded by then graduate students.

That being said the quality of a question is not only determined by its difficulty. Furthermore, in my mind, the level of a question is not only, and not even mainly, determined by its difficulty; for example, the context in which it is placed can be key to evaluate the quality of a question.

Criteria which I find relevant too include:

  • clarity of presentation
  • interestingness and relevance
  • prior effort

For example:

  • for a question to be at research-level I'd ideally expect that the presentation is in a form in which mathematics is usually written by working mathematicians (exception and accommodations can be made for users' having severe language-problems).

  • for a question to be at research-level I'd ideally expect that it is explained why the question is interesting and relevant to the asker. ("I was reading about subject S and this question Q that I find intriguing just occurred to me." can be a valid explication.)

  • for a question to be at research-level I'd ideally expect that the asker made some prior effort to answer the question by themselves; e.g., the asker checked the Wikipedia page on the subject, as questions answered on the Wikipedia page on a subject usually should not be asked on MO.

Of course, the answer should not be too obvious either, but in my mind to place too much emphasis on the mere difficulty is not the way to go. There are plenty of worthwhile question that ultimately have a rather simple answer, while there are also plenty of questions that can be very difficult but still are pretty bad questions, as they might be cryptic, unmotivated, besides the point etc.

Now, what does this mean, say, for exercises from a graduate level book or related questions. My advice would be to try to write a questions that does well on the criteria other than "difficulty" I mentioned.

It is my impression that not few users have a serious distaste for question they perceive as "lazy." Thus, something that is (or also just looks like) a sloppily copied textbook exercise or homework problem can be faced with opposition, while the essentially same question embedded into a carefully crafted post giving some context and motivation might have done just fine.

While one should not take this too far, I think it is fine that criteria other than the mere mathematical difficulty play a role in evaluating the suitability of a question; this is also common in other contexts, such as evaluations of manuscripts submitted for publication.

Finally, it is also a goal of MO to provide a place where one can somewhat casually learn about some mathematics in fields other than one's own, as a consequence of which a reasonable effort should be made that posts are accessible to a general mathematical audience (and not just a few experts in the respective subfield).

To sum this up: I think we should be open to well-written and considerate questions at a solid graduate level. By contrast we should continue to put on hold posts that seem like poorly done copy-paste jobs, ideally giving an explanation that a revised version including some context and motivation might be acceptable (if this seems to be the case).

For illustration of the above let me mention the distinct versions of the same question: the former was in my opinion rightfully closed, while I just voted to reopen the latter.

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