It sounds very much like alium triquetrum (three-cornered leek) which grows from very small white bulbs and which are impossible to dig out. I have lots of them in my bulb patch. All the top growth dies back to nothing in the summer. However, I have realised that if they appear in grass which is then mown, they don't come back. This year, when my daffodils and bluebells (and the leeks!) have completely finished, I plan to turf the area and keep it mown until next year's bulbs start growing. I'm hoping that the leek plant won't come back but if it does, I'll do the same next year!
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