States have no authority to secede; even the Confederacy recognized that. The only real check on federal power is an informed, vigilant electorate of good moral substance holding its elected leaders accountable for their fidelity to the Constitution.The only real check on federal power was the ability of the states to secede if they got fed up. Lincoln destroyed that ability when he launched his invasion of the Confederacy.
Where does it say - in the Constitution, the state Constitutions, or any law for that matter - that states don't have the right to secede? Hell, two states - Arizona and Texas - have the right to secede written into THEIR Constitutions, and refused to join the US unless that right was confirmed. Furthermore, my reading of the Constitution turns up the 10th Amendment, which says:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Since the Constitution says nothing about preventing states from seceding, or delegating to the US any power regarding secession either way, it seems to me that that power is, then, reserved to the States and the people (people can, of course, remove themselves from the United States at will).
It didn't need to say so because it's pretty obvious from the history and structure of the document. Starting with preamble's "form a more perfect Union", it's clear the intended structure of government was one of exactly that, government. Not a "league", or "alliance", or "confederation", but "Union". This "more perfect Union" was designed to replace the Articles of Confederation which begins in the opening clause with a statement of "perpetual Union".
Also, there are clauses of the Constitution that simply would not function if states have the right to secede. It would not be possible "guarantee to every State in this Union a Republican Form of Government", for example.