When you fix a single strand of rope over an edge, be it for big walls, ropes courses, caving, rescue, etc. it's often a good idea to pad the edge so tension on the rope does not damage the rope sheath.
There are lots of ways to do this. Some are some fairly DIY / spontaneous with gear you already have, such as shoving a backpack under the rope, or tying a Klemheist knot with some webbing and putting that on the loaded edge.
Another DIY rope protector that works well uses a bit of car heater hose or retired garden hose, described in this post.
You can also go with a premade product designed for this, such as the Petzl Protec (below) or something similar. This works well, and if you find yourself regularly fixing ropes, investing in specialized gear like this (about $23) is probably a good idea.
Here's another method that’s inexpensive and quick to deploy.
You probably have a section of forlorn 1 inch webbing floating around the bottom of a gearbox; now’s the time to break that out. Cut about a 2 foot section.
Slide the end of your rope inside the webbing, adjust the webbing so the middle of it is more or less over the loaded edge, and add a little athletic tape to hold it in place. That's it. (The one slight downside to this method is that you need to run the rope through the webbing before you tie any knots in the rope, so it does require a little foresight.)
Rope end is fixed around the tree with a “tensionless anchor”, and then run over the edge. A couple of feet of green 1 inch webbing protects the rope. Webbing is taped in place.