that the *Mukai-Petri map* $$\mbox{Sym}^2 H^0(C, F)\rightarrow H^0(C, \mbox{Sym}^2 F)$$ is injective, which is absurd since $3g-3<h^0(C, F)\ \bigl(h^0(C, F)+1\bigr)/2$. $\Box$

In the same spirit, we can link inequality (\[mercat1\]) to a MRC statement.

\[det7\] Let $[C]\in \cM_{g}$ be general. Mercat’s Conjecture (\[mercat1\]) for vector bundles $E$ with $h^0(C, E)=5$ is a consequence of the Maximal Rank Conjecture.

We sketch only the odd genus case, and write $g=2a+1$. From Remark \[5sections\] we know that it is enough to show that $\mathcal{BN}_C(2a+5, 5)=\emptyset$. If $E\in \cU_C(2, 2a+5)$ satisfies $h^0(C, E)=5$, then we know from Proposition \[restrquadrics\] that the image $\phi_L(C)$ induced by the determinant line bundle, lies on at least $5$ quadrics coming from the equations of the Grassmannian $G(2, 5)\subset \PP^9$. We set $r:=h^0(C, L)-1\geq 6$. Over the variety $G^r_{2a+5}(C)$ of linear series $\mathfrak g^r_{2a+5}$ there exists a morphism of vector bundles $\nu_2: \E_2\rightarrow \F_2$ which globalizes the multiplication maps $\nu_2(l)$, for $l=(L, V)\in G^r_{2a+5}(C)$. The Maximal Rank Conjecture predicts that the determinantal locus $$X_5(\nu_2):=\{l\in G^r_{2a+5}(C): \mbox{dim } \mbox{Ker }\nu_2(l)\geq 5\},$$ has expected dimension, that is, $X_5(\nu_2)=\emptyset$, hence no vector bundle $E$ with $h^0(C, E)=5$ can exist.

To close, we record the form conjecture (\[mercat1\]) takes for bundles with $6$ 