The Euphrates River remains not only a vital lifeline across much of eastern Syria, but also an important boundary area, which has become increasingly more important throughout the Syrian War.
Early in the war, Turkey warned they would not let any Kurds cross to the west of the Euphrates, and that this amounted to a “red line.” Today,US-backed Kurdish YPG forces are saying they will not allow any Syrian military forces to cross to the eastern side of the river as part of their anti-ISIS push.
The river bisects the Deir Ezzor Province, which both the YPG and Syrian military are trying to recapture from ISIS. In trying to forbid the military crossing the river, the YPG is effectively trying to claim this entire region for themselves.
If successful, the YPG would have control over nearly a third of Syria by controlling that side of the river. The Syrian government is not going to accept that, however, saying they are preparing to fight any forces they have to to recapture the entire country, including US-backed groups.
Where this leaves the US is the big question, as generally speaking the US has supported attacking Syrian government forces to help the Kurds, but officially the US also supports keeping Syria in one piece, with a strong central government, an impossibility if the Syrian government loses that much of the country.