The first method to try is bleach to see if the marks will fade that way. Coat the marks with a bleach-based spray and allow it to sit to see if the scratch marks fade.
Wipe or rinse excess bleach off. If the first attempt was successful at all, reapply until the marks are gone or no longer fade.
Wet a Magic Eraser (or generic equivalent) and squeeze out the excess water.
Gently rub over the scratches in a circular motion until they are gone. Be patient as this may take some time and persistence.
If neither of the above options works completely, purchase a cleaner called Bar Keepers Friend (found at most local superstores).
Wet the sink and sprinkle the Bar Keepers Friend anywhere there are scratches. Wet a sponge and rub in a circular motion until the scratches are gone.
If needed, rinse off and reapply as the paste gets worked into the sponge.