[For part I see ibid. 115, No. 6, 1385-1444 (1993; Zbl 0797.14016).] Let \({\mathcal F}\) be a differential field of characteristic zero with derivation \(\delta\), and let \({\mathcal C}\) be its field of constants. Assume that both fields are algebraically closed. In this paper, the author continues his studies of differential polynomial functions on schemes \(X\) over \({\mathcal F}\) and their applications to the theory of algebraic groups and in diophantine problems. The present paper studies smooth projective curves \(X\) defined over \({\mathcal F}\). The principal tools are two sequences of \(\delta\) regular maps from \(X\) to affine spaces \(\mathbb{A}^N\) over \({\mathcal F}\): the \(\delta\)-Lagrangian \(\varphi_d\) and the \(\delta\)-character map \(\psi_r\) (the dimension of the ranges are denoted \(N_d\) and \(M_r\) (respectively). -- Under the assumption that \(X\) does not descend to \({\mathcal C}\), the author proves that the \(\varphi_d\) are \(\delta\)-closed immersions (that is, come from a closed immersion \(X^\infty \mapsto (\mathbb{A}^N)^\infty\) for large \(d)\). If in addition \(X\) is nonhyperelliptic of \(\delta\)-rank \(g\) (the latter is defined to be the rank of the map \(H^0 (X, \omega) \to H^1 (X, {\mathcal O})\) defined by the cupproduct with the Kodaira-Spencer class \(\rho (\delta)\) coming from the Kodaira-Spencer map \(\rho : \text{Der}_{\mathcal C} ({\mathcal F}) \to H (X, \omega^{- 1})\), \(\omega\) being the canonical sheaf on \(X)\), then he proves that \(N_d = (g - 1) (d^2 - 1)\) for all \(d \geq 2\); that \(\varphi_2\) has finite fibres, and that \(\varphi_3\) is a \(\delta\)-closed immersion. -- Under the assumption that \(X\) is nonhyperelliptic (of genus \(g)\) which does not descend to \({\mathcal C}\), then for an integer \(r\) such that either \(r \geq g + 1\) or \(X\) has \(\delta\)-rank \(g\) and \(r \geq 2\) the author proves that \(\psi_r\) is a local \(\delta\)-immersion (that is, its \(\delta\) tangent map is injective) outside a finite set of points. This result in turn is a consequence of an infinitesimal analogue of Lang's conjecture which the author establishes. [For part III see ibid. 117, No. 1, 1-73 (1995; Zbl 0829.14020)].