The weak, the strong, the local and the global

This has been a bit quiet recently. I have been busy, working on, among other things, my first, introductory chapter. These days I have been drafting the section on physicalism (see some earlier discussion on this here). I have introduced the main notions of supervenience that are invoked on the debate on physicalism. So far, I have talked about weak, strong, local and global supervenience, characterised as follows:

(Weak supervenience): A-properties supervene on B-properties if and only if, for any individuals x and y in the actual world, if x and y are identical concerning B-properties, they are also identical concerning A-properties.

(Strong Supervenience): A-properties supervene on B-properties if and only if, if anything has property F in A, then there is at least one property G in B such that that thing has G, and necessarily everything that has G has F.

(Local Supervenience): A-properties supervene on B-properties if and only if, for any individuals x and y and any possible worlds v and w, if x at v and y at w are identical concerning B-properties, they are also identical concerning A-properties.

(Global Supervenience): A-properties supervene on B-properties if and only if, for any possible worlds v and w, if v and w are identical concerning B-properties, they are also identical concerning A-properties.

Well, there is enough food for the brain here. There has been a lot of discussion on the relation among all these notions. I am not an expert at all, but at any rate, these are my reactions (any comments or criticisms are more than welcome!). It seems to be widely held that weak supervenience is too weak for the purposes of defining physicalism, and I agree with that. Many people hold that local supervenience is stronger than global supervenience, and in particular, that there are truths that do not supervene locally on the physical, but supervene globally. I guess this depends on how you characterise the physical properties of individuals x and y (when these are the B-properties, in the schema of the definition of local supervenience above). If we allow the physical properties of x and y be broad enough, then any property P that supervenes globally will supervene locally too.
I was also wondering whether strong and local supervenience are equivalent or not. At first sight, they do seem the same thing to me, but I do not know whether this assumes something that might be controversial (you can never be too careful...). So I would like to know what other people think.

In any case, it's a relieve to think that I can be neutral on all this, for the purposes of my thesis. It does not really matter what of these notions you use to define physicalism (excluding weak supervenience, obviously): the other three resulting definitions of physicalism seem to be committed to global supervenience. And this is what conceivability arguments seek to refute. On my view, unsuccessfully. But that's another day's story...