Everyone with a PS2 has undoubtedly dealt with a scratched disc. Fortunately, if this does happen to you, there is a common household solution that is effective for repairing most minor scratches. All you need is ordinary white toothpaste, a dry cloth that is clean and preferably soft, and regular car wax, which is optional.
You should cover the entire back of the disc in toothpaste, rather than just the scratched portion. Start from the center of the disc, and then move out in a straight line. You should slowly and carefully repeat this process, until the entire back of the disc has a thin layer of toothpaste, covering it. Let the toothpaste dry for a few minutes.
Once the toothpaste is dry, take the disc to the sink, and gently rinse it off, using your fingers again. Be sure to rinse every last bit of the toothpaste off, while trying not to be too forceful with your motions.
Be extremely gentle as you dry the disc off using a soft clean cloth. If you would like, you can apply some car wax to the back of the disc, as it can protect it from getting dirty in the future.