So I was thinking, what if I had ac mains converted to dc power (transformed, rectified, and smoothed) ran the dc through a mosfet controled by the arduino (so now I have the dc variable power) and then ran that through an inverter. So the inverter output will be variable because it is getting variable dc input from the mosfet.
Full-wave rectified AC through the PWM-driven MOSFET would probably work. There should be no need for "transforming", "smoothing", or "inverting". (You will loose a couple of volts across the full-wave rectifier.)
However, you do need to isolate the MOSFET (and the other high voltage stuff) from the Arduino (and yourself!) with an opto-isolator. And remember, the peak AC voltage is about 1.4 times the RMS voltage. So make sure your MOSFET and the high-voltage side (output-side) of your opto-isolator can withstand the applied voltage.
I have made a 2-position dimmer switch for regular 120VAC incandescent lamps with a half-wave rectifier, but I've never tried making a full-range PWM "AC" dimmer. I don't remember if I've ever connected a lamp to a full-wave rectifier, but there's no reason it won't work. Of course, these ideas will NOT work with fluorescent lights, or with most AC LED lights which actually require AC (but a regular phase-controlled dimmer wont work with these either).