After the flower blooms and fades, the sprig gets a long row of little bumpy seed like things. Are these the corms that need to be planted (corms?) or do you divide the actual bulb from below the ground and this forms new plants? Do the bulbs or corms both blossom the next year?
You can cut back the seed heads and stalks, but it is best not to cut back the leaves until they die on their own at the end of the growing season. The plant needs them to store energy for growing and blooming next year.