Yes, any EC based crypto needs to choose a curve, but a point on a curve (like a public key `xP`) is meaningless without knowing which curve it lays on (unless you're mounting invalid curve attacks, in which case you're taking advantage of a peer maybe ignoring that), just like your public DH key (`g**x`) is meaningless without knowing which prime parameter it was computed against. So, just like the actual value of the pubkey, the parameters would be part of the certificate.