Wash the cooktop with a mild detergent and water on a sponge and degreaser as necessary. Ensure the cooktop is free of any debris or grease before sanding the top.
Dry the cooktop completely with a towel.
Rub the scratched area firmly with medium-grit sandpaper, such as 240-grit. Rub it back and forth going with the length of the scratch -- not across it.
Wipe any dust away from the surface and repeat with a finer grit of sandpaper. Wipe away the rest of the dust and then use a nylon scrub pad with a polishing agent to continue to buff out the scratch.
Use a finer buffing pad to polish the area completely.
If the black finish came off, repaint the area with a high-gloss enamel or appliance paint. These are usually available in spray cans at hardware stores.