Basic Slayer Exciter
The rest of the circuit makes sense, however I am wondering if this wire can be disconnected from the circuit. If a charge is being accumulated at the "275", where it will ultimately discharge as an arc, why is it necessary to connect this wire? Would it be unnecessary if it was an AC current source?
That circuit relies on the wire in question to create oscillations to drive the transformer.
Without that wire the transistor would never switch off. There would always be DC current flowing through the primary side of the transformer. Since the current wouldn't change, there would be no coupling to the secondary (high-voltage) side.
The LED's primary purpose is to make sure that the transistor base isn't pulled too far under 0 volts. A standard diode would also work, but this way you get an visible indicator for free :)
There is a great answer here that goes into more detail about why the circuit works.
No, you can't remove that connection. That is the feedback that closes the loop that makes the whole thing oscillate.
In fact, this circuit relies on some amount of capacitance or leakage or transmission current from the open end of the secondary. If that open end were truly open, then the feedback would have no reference and would do nothing. Perhaps even the parasitic capacitance of the transformer is enough to provide the necessary coupling, but one way or another, current has to flow thru the point you marked, which means some coupling back to circuit ground or power has to exist at the open end of the secondary.
