Well, what it says on the can really, what if one of the colonial powers claimed Antartica, even as one giant whaling station, but what happens if they do.
Why they claim it, I'll leave up to you, who claims it, well I'd argue towards France, the United States, or Britain since they were the earliest countries to land expiditions in the Antartic. If France claimed it though, then it might be taken from them after the Revolutionary wars, which although Napoleon might be butterflied away, would probably still happen, the King was incompetent after all.
So... go wild.