As Izuku gain the quirk(not really) of scarf everyone seen that it was weak but it literally contained all of his emotions and powers and mostly because no one knows what happens and mostly the scarf is one of the most powerful and only Izuku can handle it as it weights (My Au) is weighted 85 million grand that no one was able to lift up or touch it while it's in Izuku body. Izuku only takes it off when he takes a shower and mostly he contains how much it weights even his own family hated him because of the scarf and they abuse and neglect him because of the scarf as they believe that it's weak. There's only 1 person who doesn't believe that it's weak because she read that this scarf is the legendary quirk and that no one would ever take it not even her and it was Momo and she mostly told her family about it and they went to go look for him until she saw him in Ua.
Ship:
Izuku x Momo
Izuku will always wear the black scarf unless he's taking a shower