Question: Suppose you're working on a lengthy text that you would like to publish as a book or a "research monograph". For example in mathematics stuff like Springer Lecture Notes, Memoirs of the AMS, or any other book series.

Then say you upload a preliminary version of the book somewhere, say arXiv (with the usual non-exclusive arXiv license), or to your personal website. Is there any issue with this, if later you want to submit to a publisher? I am wondering how common it would that a publisher would desk reject your book, if it is already on the arXiv. Most of the publishers do not seem to provide this kind of information very clearly.

And if later you publish, do the agreements with the publisher usually allow you to update your arXiv draft with corrections?

Of course it is best to confirm directly with the publisher that one wants to work with. But I would be interested in any general advice or anecdotes on these issues, especially in the fields of mathematics and physics. I have seen several authors who put all their papers on arXiv, but do not upload books/monographs.

Answer: 
I "published" versions of my textbook on my personal website (with an explicit copyright notice) so that students and some faculty elsewhere could use it when it was under development. The feedback was very valuable. No potential publisher objected to this, even after the book was under contract.

When the published book appeared the publisher naturally wanted those versions deleted. I still sometimes see them posted elsewhere, and ask for deletions (same for pirated copies of the actual published text).

I think arXiv would be a mistake. You can't simply "delete" there. Nor would it be the right place to post updates and corrections. That's your publisher's call.
Every publisher has its own standards on whether it will be willing to consider publishing something that has already appeared in some form, including ArXiv. You will need to check their websites and/or editors to see what is acceptable. I suspect that for books, they tend to be a bit more conservative than for articles, since they normally have a larger investment in the production process (copy editors, for example).

But you should also consider, as any publisher will, whether publishing the work, largely complete, will kill most (all?) future sales. I'd be especially careful about publishing complete exercise sets for any text books as those exercises are often the most important part, unless the topic is some advanced one that has few competitors. The latter may be your situation, of course.

But self publishing (ArXiv or other) will limit your choices, so ask before you jump. It is useful in any case to talk to one or more editors about your manuscript to explore their interest.