Extrinsic projective curves X¹ in P²(C): harmony with intrinsic cohomology

On a geometrically smooth complex algebraic curve X^1 in P^2(C), represented in complex affine coordinates (x,y) as the zero-locus R(x,y) = 0 of some polynomial R of degree d >= k+3, an explicit family of generating independent holomorphic jet differentials J_R^1, ..., J_R^k expressed in terms of R and its partial derivatives is exhibited with its new precious nonlinearity features as a complete explicitation of all holomorphic sections of the Green-Griffiths bundle of m-homogeneous polynomialized order k jets of local holomorphic maps from a complex disc into X^1.