A rule to live by: I won’t use anything I can’t explain in five minutes. (Philip Crosby)
Every now and then as editor, I see an author upset at a rejection of a paper because the referee “clearly did not grasp the key point of the paper”.
In many cases this is because the key point is not stated prominently enough in the introduction, instead being buried in a footnote, an obscure remark, a lemma, or even not explicitly mentioned at all.
This can be as much the fault of the author as it is of the referee; it is incumbent on the author to state as clearly as possible what the merits, novelties, and ramifications of the paper are, and the fact that an expert in the field could read the introduction and not see these is a sign that the introduction is not yet of publication quality.
In particular, the introduction should spend some time comparing and contrasting the paper to other literature, and demonstrate why the paper’s results and techniques are new, interesting, and/or surprising given this context.For instance, if new difficulties had to be resolved here which were not present in previous work, or if counterexamples indicate that the result or proof cannot be improved in various obvious directions (e.g. by dropping a hypothesis, strengthening a conclusion, or by using a simpler method in the literature), then these points need to be made prominently.
The introduction should also clearly state (or at least paraphrase) the main results of the paper, and ideally should also outline how and where these results are to be proved. Of course, these results need to be described accurately and in appropriate detail when doing so. If the main result is too technical to be placed in the introduction (e.g. because it requires a large number of complicated definitions in order to state precisely), then a simpler (but still interesting) special case can be stated instead, or an informal statement of the theorem given (but in the latter case it should be clearly indicated that that this statement is informal, and that a more precise version will be given later).
For similar reasons, the title and abstract should get right to the point and make it clear what the substance and novelty of the paper is; remember that these are the first impressions that the reader will have of your paper, and so you should make the most of that opportunity.
See also “Organise the paper“.