After the piece has cooled and been removed from its ring, you can consider how to finish the piece. If you have used something other than a flat rim, you may have minimal work to finish the piece.
As most people use a flat ring, the first decision is whether to retain or remove the rim from the vessel. In some cases the rim can be retained as an integral part of the piece. Again here there is little work needed to finish the piece.
For most aperture drops and for most people, it is desirable to remove the rim. To have successful drops without rims, you most often need to have access to cutting and polishing equipment.
You can use a tile saw or band saw to cut off sections of the flat rim and then a linisher to grind the edges to round, followed by polishing. This will give you a thick rim.
If you want a thin rim, you will need to cut through the drop at the top - visualise a cut at right angles to the length of the drop. Usually tile saws are too aggressive for this. If you can find a band saw with a high enough clearance, you could gently separate the rim from the drop after having reduced the size of the rim to make the use of the band saw more easy.
In both cases you must grind and polish the edge of the rim to give a finished appearance. Fire polishing is not possible as the drop would collapse long before the rim was smooth.