The Secp256k1 curve is used in cryptocurrency. Can someone generate a private key with a different curve?
The Bitcoin network expects secp256k1 when adding something to its blockchain. If you use another curve, it naturally will not produce a valid secp256k1 public key capable of validating the secp256k1 signatures and will be rejected by the network. If some client uses a bugged or even backdoored version of the curve, the client will simply not be able to add anything to the blockchain. This would only become a risk if a significant portion of the network ended up using this same bad library, effectively resulting in a fork of the core client and an entirely new, separate protocol.