Polus' task seems a bit easier this go around - research in the Mystic Archives. The Council of Mages is tied up with their researches on Necromancers and has been getting nowhere, but perhaps the problem is that the questions they are posing to the librarian are too broad. Maybe something a bit more specific will yield some results. Polus want you to hop over to the Archives and talk to Tar-Meena about those Black Soul Gems that you found in Falcar's quarters.
Tar-Meena is obviously overworked, but is happy that you have something more specific than "tell me all about Necromancers." She suggests that you read "Necromancer's Moon," which is laying around here somewhere. In fact, it's right behind you on the little table in the corner. Take it and read it. After you have read it and get the appropriate journal update, talk to Tar-Meena again. She's not sure about the "shade of the Revenant" part, but lets you borrow the book. Be sure to remind Polus to bring it back.
Polus isn't too sure on the reference, either, but suggests that you talk to Bothiel, the keeper of the Orrery (inaccessible unless you've purchased the official plug-in). She's probably wandering around Polus' chamber, too, but may be out on the University grounds somewhere. There's something familiar about all of this, as Bothiel will tell you. In fact, Falcar was researching much the same topic a little while back and had a file full of notes on it. As it just so happens, he dropped some of his notes and Bothiel picked it up and passes you the "Hastily Scrawled Note." The note mentions four locations where it says that altars have been raised: Dark Fissure, Fort Istirus, Fort Linchal, and Wendelbek.
Polus only recognizes one of the locations, Dark Fissure, which is in the mountains near Cheydinhal. He marks it on your map. Dark Fissure is quite a haul, though, and somewhat difficult to get to because it's up on the side of a mountain. Fort Linchal is northwest of Kvatch; Fort Istirus is south of the road, about halfway between Skingrad and Kvatch; Wendelbek is about halfway between the two north-running branches of the Panther River, kind of east-southeast of Bravil.
Which one to use? Well there are good things and bad things about all of them. As I already mentioned, Dark Fissure is a bit difficult to get to. But you'll have greater odds of getting what you need, quickly, if you go there. The two Forts are a lot closer to home and are also close enough to each other that if you don't find what you need at one, you can hoof it to the other fairly easily. I'd go with Wendelbek, though. Why? Greed, pure and simple.
Wendelbek is a multi-level Ayleid ruin inhabited by several Necromancers. You should be able to pick up a couple or three dozen Welkynd Stones and three Varla Stones from the ruins themselves. Necromancers are a great source for healing and sorcery potions and destruction-based scrolls. At least a couple of the Necromancers should have enchanted staves. And, depending on your level, some decent weapons are very probable. Additionally, there are a lot of Ayleid Reliquaries and Coffers about, at least one complete set of Novice-level alchemical gear per level and a good handful of Lesser and Petty Soul Gems. All in all, you can get quite a haul from Wendelbek. And if you don't arrive at the proper time, it gives you something to do while waiting for the Worm Anchorite to appear.
Your job is to locate a Worm Anchorite near the altars outside of any of these locations and watch him create some Black Soul Gems. The Worm Anchorite is only there at night, one day per week and the day is different for each location. If the weather is in your favor, this will be really simple because what you're looking for is a shaft of light shining on the altar (the light is kind of hard to see when it's raining). This only happens at night, one night per week. When you see the light, the Worm Anchorite should be there. With a good Stealth skill, you can watch him do his thing and be done with it. It not, you'll have to take him out. He'll have some instructions in his inventory and some Black Soul Gems will either be in his inventory or in the altar.
Any one of three things will complete this quest for you:
Then report back to Polus. He'll promote you to Magician and give you some nifty robes. This is also his last quest. From here on out, you'll be reporting to Hanibal Traven, the Arch-Mage. Although Traven will be pretty good about reminding you to do it, you should talk to Polus after completing each of the later missions because he's still the guy who handles your promotions.
Now for a little history. In the Elder Scrolls games, soul gems are used to create and/or recharge magical items. In previous games you could only soul-trap creatures, never NPCs. Not so in Oblivion. Black Soul Gems are what you use to Soul Trap NPCs. From a game perspective, there is absolutlely no difference between what you can do with a Grand Soul Gem and a Black Soul Gem. There are no special benefits to be gained by Soul Trapping NPCs. They're just Grand souls as far as the game is concerned, so this is more of a role-playing tool than anything else. The one difference is that you will generally not be running into "Grand" souls until you have leveled up to about the teens. With a Black Soul Gem and a Soul Trap spell, however, any old Bandit will do.
You have to have a Black Soul Gem in order to get Vaermina's quest for one of the Daedric Articfacts and Polus won't take the ones you retrieved on this mission from you. You don't actually have to hold onto one as you'll probably encounter them as leveled loot in Necromancer Lairs. Vaermina doesn't care whether it's filled or empty, though. So if you're level 5 or higher, you're now set for that quest.