Poppies always make me think of
In Flanders Fields...
Try a quick image search for
damping off. If the stem is discolored and thread-like, it may be that they have this fungal infection. Perhaps too much water lately?
Otherwise, if it is in a greenhouse, perhaps they could benefit from a little more breeze to become sturdier. Plants need a little stress in their life to grow stronger, just like people.