for a *general* curve $[C]\in \cM_g$ remains a very stimulating one, and which can be naturally connected to the *Maximal Rank Conjecture* (MRC) in the form that appears in [@AF].

The original version of the MRC is due to Harris [@H] p. 79, and it amounts to the following: Let $C\subset \PP^r$ be a smooth curve of genus g and $\mbox{deg}(C)=d$, corresponding to a general point of the unique component of the Hilbert scheme ${\bf{Hilb}}_{d, g, r}$ mapping dominantly onto $\cM_g$ (that is, in the range $\rho(g, r, d)\geq 0$). Then the restriction maps $$\nu_m(C):H^0(\PP^r, \OO_{\PP^r}(m))\rightarrow H^0(C, \OO_C(m))$$ have maximal rank. In particular the Hilbert function of $C$ is minimal. One can generalize Harris’ Conjecture in two directions: Either $(a)$ by requiring that $[C]\in \cM_g$ be *general in moduli* rather than in the Hilbert scheme, then conjecturing that the restriction maps to $C$ be of maximal rank with respect to *all linear series* of type $\mathfrak g^r_d$, or $(b)$ by asking for the minimality not only of the Hilbert function but of the entire *graded Betti diagram* of $C$ (see Section 5 for how such a prediction can be correctly formulated). The generalization of Harris’ Conjecture in direction $(a)$ was 