$l:=(L, V)\in G^3_d(C)$ and use the elementary fact that if $\mbox{Ker } \nu_2(l)\neq 0$, then there exist pencils $A_1, A_2$ on $C$ such that $L=A_1\otimes A_2$. By Brill-Noether theory, $\mbox{deg}(A_i)\geq [(g+3)/2]$ for $i=1, 2$, hence $\mbox{deg}(L) \geq g+2$, which is a contradiction.

Mercat’s conjecture
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We follow standard notation and denote by $\cU_C^s(n, d)$ (respectively $\cU_C(n, d)$) the moduli space of stable (respectively semistable) vector bundles of rank $n$ and degree $d$ on $C$. If $L\in \mathrm{Pic}^d(C)$ is a line bundle, we set $\mathcal{SU}_C(n, L):=\{E\in \mathcal{U}_C(n, d): \mbox{det}(E)=L\}$ and $\mathcal{SU}_C^s(n, L):=\mathcal{SU}_C(n,L)\cap \cU_C^s(n, d)$.

Recently, Lange and Newstead [@LN] proposed a definition of the Clifford index of a higher rank vector bundle. For $E\in \cU_C(n, d)$, the *Clifford index* of $E$ is the quantity $$\gamma(E):=2+\mu(E)-\frac{2}{n}h^0(C, E)\geq 0.$$ By Serre duality, $\gamma(K_C\otimes E^{\vee})=\gamma(E)$. The *higher Clifford indices* of $C$ are defined as $$\gamma_n(C):=\mbox{min}\bigl\{\gamma(E): E\in \cU_C(n, d), \ \ \mu(E)\leq g-1, \ \ h^0(C, E)\geq 2n\bigr\}.$$ Note that $\gamma_1(C)=\mathrm{Cliff}(C)$ is the classical Clifford index of $C$. Several foundational properties of the invariants $\gamma_n(C)$ are studied in [@LN]. For instance the following inequality follows from the definition and is implicitly used in loc. cit.

$\gamma_2(C)\leq \mathrm{Cliff}(C)$.

We choose a line bundle $A$ on $C$ 