The remains of the old leaves can look thicker and finer than in the picture. In the picture the plant had been underground dormant when I dug it up (spring sickness investigation), and I had washed it to get a good look at everything, so the fibers are wet changing the appearance, but it was the only pic I could find that showed them. You mention it is around the stem/stalks. Do you mean it is only around the flower scapes? I agree, a picture would definitely help.
