paper.

The Maximal Rank Conjecture
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In [@AF] a strong version of the *Maximal Rank Conjecture* (MRC) for general curves has been formulated and its various applications to the birational geometry of $\mm_g$ have been presented. Since MRC will turn out to be also connected to rank two Brill-Noether theory, we begin by recalling, in a somewhat restricted form, the set-up from [@AF] Section 5.

We fix positive integers $g, r, d$ such that $\rho(g, r, d)\geq 0$, as well as a general curve $[C]\in \cM_g$. We may assume that $G^r_d(C)$ is smooth of dimension $\rho(g, r, d)$. For a linear series $l=(L, V)\in G^r_d(C)$ we denote by $$\nu_2(l): \mathrm{Sym}^2 (V) \rightarrow H^0(C, L^{\otimes 2})$$ the multiplication map at the level of global sections. After choosing a Poincaré bundle on $C\times \mbox{Pic}^d(C)$, following [@ACGH] Chapter VII, one can construct vector bundles $\E_2$ and $\F_2$ over $G^r_d(C)$ with $\mbox{rank}(\E_2)={r+2\choose 2}$ and $\mbox{rank}(\F_2)=h^0(C, L^{\otimes
2})=2d+1-g$, together with a bundle morphism $\nu_2:\E_2\rightarrow \F_2$, such that for $l\in G^r_d(C)$ we have that $$\E_2(l)=\mathrm{Sym}^2 (V) \ \mbox{ and } \F_2(l)=H^0(C, L^{\otimes 2}),$$ and $\nu_2(l)$ is the multiplication map considered above. Since $[C]\in \cM_g$ satisfies Petri’s theorem, $H^1(C, L^{\otimes 2})=0$, therefore by Grauert’s theorem, $\F_2$ is 