I just got Invisalign today and there is a very sharp edge that is cutting my mouth. What should I do?

You have two options:Go back to your orthodontist and ask them to smooth off the sharp area or smooth it off yourself with an emery board or nail fine (gently!).

You can easily trim a sharp edge off your aligner, and I suggest that you trim it with a fine curved scissor or a nail clipper, not a file.  A file tends to leave a rough edge and you're much better off with a clean cut.  By the way; just a few millimeters, not too much, will do it.  To be safe, why not call your orthodontist to be sure it's OK with him/her?

The edges of the Invisalign tray are often irregular or jagged and can cause the irritation that you describe.  Your dentist/orthodontist can easily smooth and polish the edge, but in the meantime an emory board or nail file can be used to soften the edge at home until you are able to see your dentist.