I've written a lot about this and had many discussion with both customers and colleagues and it gets very involved.... however:
The short answer is that the DM is there to provide broad planning and logistical support (emergency equipment, spare weights, food and water etc.) within an environment appropriate to the dive team; act as a guide when necessary and a supervisor when not. All divers in the water are certified divers and therefore should be able to look after their own safety and follow the plan as laid out by the DM.
Technically speaking, that's about it. If a diver gets separated from the group, that's their problem, not the inattentiveness of the DM. If a diver runs out of air, that's their problem. Corks to the surface, nope, nothing to do with the DM.... BUT - after guiding several thousand dives, what I can tell you is that many divers expect to do absolutely everything. One diver complained about me because I did an air check at around 25 metres and he reported he had only 30 bar. 15 minutes into a dive. I stuck him on my octopus, swam him back to the reef with 7 other divers following, surfaced him, sent him back to the boat, then went back down and finished the rest of the 60 minute dive. I said he couldn't dive again without a private guide or a refresher course. He refused and complained about me because I should have looked behind me more often and checked air more regularly. It was all my fault.
That's just one example but it's something I became very wary of - if you read my blogs "what is a dive guide" - you'll see some of the techniques I used for checking the dive team without actually looking like I was checking the dive team - because one of the problems in this very litigious day and age is that it doesn't actually matter what the law is or your responsibilities are, if somebody sues, you're in the poopoo, even if you didn't do anything wrong.
For that reason, I basically told myself I was to assume I had total responsibility for all the divers under my supervision, even though in reality I didn't. It's a constant source of debate and ridicule on SB about how many poorly trained divers are out there - and it's true, there are, I've dived with a bunch of them. It gets to be a fine art after a while - solid control without being authoritarian and waving your stick at everybody underwater! On the other hand, there were many days when I would have a group of experienced divers behind me who listened to the plan, did everything that was asked of them, and if they disappeared from view every now and then, this was okay because I had dived with them many times and knew they would catch up. On days like that all I had to do was find cool stuff which - given I dived the same reefs on a daily basis I knew where most of that was - and make sure we got home on time.
It's complicated; more so than many would have you believe. Agencies give you the rules, many divers expect more. Balancing the roles and responsibilities of a divemaster between agency standards, legal requirements and customer experience is - really - a complex challenge. Especially if you can't find any cool stuff - then you're *really* in the doodoo!!
Cheers
C