i've some trouble understanding the base of the range proof presented at https://elementsproject.org/elements/confidential-transactions/investigation.html
I've understand the base of the Pedersen Commitment but I lost when he speak about commitment to zero:
A pedersen commitment can be proven to be a commitment to a zero by just signing a hash of the commitment with the commitment as the public key. Using the public key in the signature is required to prevent setting the signature to arbitrary values and solving for the commitment. The private key used for the signature is just the blinding factor.
Why to prove a zero commitment it make an hash of the commitment? Does he intend this ( if we call $C$ the commitment)?
$Hash(C)H + C$
But if I know the Publik key $P=kG$ cannot simply check if $C= P$ because we have that $C= kG + 0H$?