Well the French COULD have moved further north from Syria than they did and created a Kurdish mandate in Western Kurdestan centered around Diyabakir, maybe extending all the way to Van in 1920. Interestingly, there are three major oilfields around Diyabakir too, as it turns out, one of which was first drilled in 1939, so it was fairly obvious.
Of course the French would have been under pressure by Arab nationalists to consolidate French Kurdestan into a Greater Syria under Sunni Arab rule as in OTL. But with a Kurdish population in such a Greater Syria at 40%, when non- Sunni Arabs would be taken into account, Sunni Arabs would make up only an equivalent 40% of the population with the balance of the population held by non-Sunni Arabs such as Alawites, Christians and Druse. Altogether a very different place, much more resembling Iraq than the Syria we know OTL. That's assuming that the French did not simply give the Kurds an independent state with acces to the Mediterannean via Hatay and Alexandretta.