Question
What can I do to rid my variegated schefflera of brown leaf spots which start off in the center of leaf and when touched fall off.

Answer
Could be a variety of problems from bacterial and fungal disease to environmental issues.  I would remove any affected leaves and treat with a copper based fungicide.  Hi-Yield brand sells a product called copper fungicide.  Most garden centers have this product.  The copper has activity against both fungi and bacteria, so we can attempt control of each with one product.  Mix according to label instructions for ornamentals and spray the plant outside in the shade.  The product is safe and considered organic.  Lets try this first and go from there.  Let me know what happens in a couple of weeks.