  We extend the $\hat U$-theorem for projective varieties proved in
arXiv:1601.00340 and arXiv:1607.04181 to projective schemes over $\mathbb C$.
The condition "ss=s" required in arXiv:1601.00340 and arXiv:1607.04181 is
reformulated algebraically for our setting. We also describe a blowing up for
the condition "ss=s". Different from the sequence of blowing ups in
arXiv:1607.04181 , we can get "ss=s" in one blowing up.
