Here, I'll give it a shot:
DoC showed that not all the people with Jenova's cells decided to go have them removed. They may not have even understood why they had Geostigma in the first place.
So we have people all over the world still living with small fragments of Jenova inside their bodies (see, I'm even throwing the word "fragments" in there so it kind of sounds like the stuff they did in DoC).
Because Lucrecia's cave is where Chaos and the tainted mako were discovered, it also made for a great place for what remained of Sephiroth's Lifestream to take up residence, hiding out until he grew strong enough to act through it again.
So, the remaining Jenova cells scattered through all those people are gradually brought together in the body of one Lucrecia Crescent, the person with the most Jenova material still inside her body. Maybe this happens because Sephiroth's will manipulates someone into it, or maybe he takes over her body. Isn't really important right now.
At the same time that all this is going on, we learn that it was either Chaos' full awakening or the destruction of Omega Weapon near the end of DoC that awakened Genesis. Because of the prior loss of the rest of the Weapons, Minerva accepted him as the planet's new guardian, to awaken in its time of need, and granted him all the power of a Weapon during his sleep.
Since he didn't wake up when Meteor was summoned, he probably sealed himself away after it happened, shortly before the Tsviets defeated the Restrictors only to be trapped during Meteorfall.
Now awake, Genesis senses the growing danger to the planet as Sephiroth tries to revive. Also, because he's hanging out with Weiss, whose body Nero merged with, he knows about tainted Lifestream, which is connected to Lucrecia, who's connected to Shelke.
Believing that Shelke could be useful in locating or hindering Sephiroth, who's gathering/acting through Lucrecia's body, he kidnaps her, bringing him into conflict with Cloud and the rest of the team.
Various things happen, but Genesis and the others are unable to prevent Sephiroth's eventual rebirth, which takes place in the body of his own mother.
Genesis decides that the only way left to save the planet is to try bringing back Zirconiade. Elfe and the Turks obviously oppose him, as does Avalanche.
Things culminate in a three-way battle between all factions (maybe even a three-way battle between Cloud, Genesis and Sephiroth, thematically tying things back to the three-way fight in CC between Angeal, Genesis and Sephiroth), with Sephiroth and Genesis being defeated for good.