Nutsedge has a small bulb or "nut" underground that I'd expect makes it very resistant to something like roundup that kills foliage. Don't know what you'd use to kill it, though, since the most popular products I know aren't safe on northern grasses like fescue (IIRC).

Also, cutting your grass high can be good but most warm-season grasses will develop bad thatch if not cut regularly. The stems will grow on top of each other, then all die over the winter because they're not in the ground and are exposed to the elements. Then the stems (which are like wood rather than leaves like grass blades) will not decay fast enough in the spring, so the new stems will grow on top of them and you'll repeat the cycle. With grasses that need it (centipede, bermuda, etc), cut them low, or else...