The easiest is to fill the scratches with an acrylic scratch remover, which is applied as a liquid and dries hard and transparent, hiding the damage by filling the scratches chemically. For deeper scratches, the most popular fix is buffing the glass with cerium oxide, a popular glass, ceramic and metal polish with very low abrasiveness—it's widely used in jewelry and ceramics. Add water to the fine powder and make a slurry the consistency of Elmer's Glue, then cover the damaged areas with the paste. Using a drill with a hard rubber polishing wheel attached, apply firm pressure to slowly grind the surface smooth. To make things a bit easier, mark the location of the scratch on the other side of the glass so you can keep track of where you should be working. It may take a while, but keep the paste moist and add more cerium oxide as needed, and the result will be a smooth, scratch-free surface.
