The Scherk-Schwarz compactification is just an extremely special kind of compactification of one dimension in which the spacetime fermions are chosen antiperiodic along one circle of the compactification manifold. It's extremely far from a "general compactification" of string/M-theory.

General compactifications of string/M-theory allow many features that can't be discussed by the simple formulae above; in some sense, all of string/M-theory is needed to answer the question what the general compactification can be and cannot be. General compactifications may include a manifold. It doesn't have to be a manifold; it may be an orbifold with orbifols singularities. In fact, orbifold singularities are not the only allowed ones; one may have conifold singularities and probably much more general ones, too. Various fields may have nontrivial monodromies. One may wrap branes, try to incorporate boundaries at the "end of the world", and add many kinds of Wilson lines, fluxes, generalizing the electromagnetic fluxes, with various quantization conditions, and other features, some of which may even be unknown at the present moment.

The compactifications may get strongly coupled at various loci and interpolate between totally different descriptions such as type II string theory and M-theory, too. Moreover, one may have non-geometric compactifications, too. The question as formulated seems to be too broad.

Moreover, I also have to disagree with the first comment written under the question. It's a string/M-compactification, not a dimensional reduction, which is a consistent theory. The simple dimensional reduction is just an approximate effective theory at distances much longer than the size of the compactification manifold and such an effective theory almost always suffers from some kind of an inconsistency. To fix these inconsistencies, one needs to consider the fully consistent theory, namely the string/M-compactification.

There is no "universal algorithm" to "upgrade" a dimensionally reduced theory to a compactification (the term "upgrade" may mean either "dimensional oxidation" if we just try to add dimensions to a theory; or "UV completion" which means finding the precise compactification including all the short-distance physics that may be approximated by a given effective theory). Many compactifications – as many as the notorious number $10^{500}$ – of compactifications may lead to very similar effective theories in the large dimensions. There's no easy way to find the "correct one" among them.

It's also hard to understand in what sense "most of literature" discusses the dimensional reductions rather than compactifications. I don't think it's the case. Of course, if one looks at literature about dimensional reductions only, one may get this conclusion. However, true literature on string theory doesn't agree with the statement. It's mostly about the full physics of compactifications, not just the reductions – otherwise it wouldn't really be a stringy literature. This is true pretty much by definition.

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