What Does a Mole That Comes Back with a Comment 'The Peripheral and Deep Margins Are Positive' Mean?

The simple answer to this question is that this mole must be re-excised.  Moderate to severe cytologic atypia means that the nuclei of these cells are unusual and are following a progression toward malignancy.  Positive margins means that some of the mole remains in your body.  In pigmented moles, this generally relates to the appearance of the melanocytes, or pigment producing cells.  Moles with moderate to severe cytologic atypia have a higher chance than normal moles of transforming to melanoma and must be removed (surgically).

It means the margins aren't clear and they are "positive" for the atypical notation cited. When you have a removal done like this, you want the margins to be clear, so you need to have further excision or shave biopsy or currette of the area done to completely remove the affected area.