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You can sue for a facial scar if it is the result of negligence on someone's part. There may also be other bases for a lawsuit. How much a case will be worth and whether it is worth suing depends on how you got the scar and who was involved.
If you went to see a plastic surgeon and the doctor left you with a visible unwanted incision, your case is likely worth a lot of money. This is all the more true if you happen to rely on your face to make money, and that applies to more than just models. But if your scar is the result of a barroom brawl and the perpetrator is in prison, say, it may not be worth your while to sue.
Bases for a Case
Let's consider bases for a case using the plastic surgery example above. You could sue for medical malpractice and, possibly, breach of contract, depending on the details of the deal you struck. If your surgeon misrepresented credentials or made other false statements to induce you to sign up for services, you may also claim misrepresentation.
In a medical malpractice case you may also sue the hospital where the surgery was done. There may be an equipment malfunction to blame in some cases. An attorney will be able to tell you what claims are viable depending on what happened.
Now let's look at the barroom brawl case. While your face is no less precious than that of a person scarred by a surgeon, bases for a suit are probably fewer.
Perhaps you can sue the bar for negligence in failing to maintain a safe establishment. Perhaps you got lucky (relatively speaking) and were slashed by a trust fund kid. Then, you may want to sue for damages for your facial scar even if the person responsible is incarcerated or unemployed.
Have Your Case Assessed
The devil is famously in the details and cases cannot be assessed in the abstract. If you have a facial scar and believe someone is responsible for it, speak to an attorney. Tell your story and find out more about how that person may pay for what happened.