Linear stability and stability of Lazarsfeld-Mukai bundles
Let $C$ be a smooth irreducible projective curve and let $(L,H^0(C,L))$ be a complete and generated linear series on $C$. Denote by $M_L$ the kernel of the evaluation map $H^0(C,L)\otimes\mathcal O_C\to L$. The exact sequence $0\to M_L\to H^0(C,L)\otimes\mathcal O_C\to L\to 0$ fits into a commutative diagram that we call the Butler's diagram. This diagram induces in a natural way a multiplication map on global sections $m_W: W^{\vee}\otimes H^0(K_C)\to H^0(S^{\vee}\otimes K_C)$, where $W\subseteq H^0(C,L)$ is a subspace and $S^{\vee}$ is the dual of a subbundle $S\subset M_L$. When the subbundle $S$ is a stable bundle, we show that the map $m_W$ is surjective. When $C$ is a Brill-Noether general curve, we use the surjectivity of $m_W$ to give another proof on the semistability of $M_L$, moreover we fill up a gap of an incomplete argument by Butler: With the surjectivity of $m_W$ we give conditions to determinate the stability of $M_L$, and such conditions implies the well known stability conditions for $M_L$ stated precisely by Butler. Finally we obtain the equivalence between the stability of $M_L$ and the linear stability of $(L,H^0(L))$ on $γ$-gonal curves.