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I'm trying to understand a passage from an article by R.Lazarsfeld concerning properties of the Lazarsfeld-Mukai bundle for my reseach. Here is the paper of Lazarsfeld giving the construction (p.3 of the pdf) : R. Lazarsfeld - Brill-Noether-Petri without degenerations. I’m paraphrasing the beginning of construction.

Here, $X$ is a complex projective $K3$ surface and $C_0 \subset X$ is a smooth irreducible curve of genus $g$. Given a curve $C$ and integers $r,d$, we define $V^r_d(C) \subset \text{Pic}^d(C)$ to be the open subset of $W^r_d(C)$ (this is the variety that parametrizes complete linear series of degree $d$ and of dimension at least $r$ on $C$) consisting of line bundles $A$ on $C$ such that :

  1. $h^0(A) = r+1$ and $\deg(A) = d$ ;
  2. both $A$ and $\omega_C \otimes A^*$ are generated by their global sections.

Fix now a smooth curve $C \subset X$ in the linear series $|C_0|$ and a line bundle $A \in V^r_d(C)$. We associate to the pair $(C,A)$ a vector bundle $F_{C,A}$ on $X$ of rank $r+1$ as follows. Thinking of $A$ as a sheaf on $X$, there is a canonical surjective evaluation map $$H^0(X,A) \otimes \mathcal O_X \xrightarrow{ev_{X,A}} A$$ Take $F_{X,A} := \ker ev_{X,A}$ ; this is the desired vector bundle of rank $r+1$. Setting $F = F_{X,A}$ for simplicity, let $E = F^*$, this is what we call the «Lazarsfeld-Mukai bundle». On has by construction the exact sequence $$(\Delta) ~~~~~~ 0 \to F \to H^0(X,A) \otimes \mathcal O_X \to A \to 0$$ of sheaves on $X$. This is the passage that I don't understand : dualizing $(\Delta)$ gives $$0 \to H^0(X,A)^*\otimes \mathcal O_X \to E \to \omega_C \otimes A^* \to 0$$

I don't understand how we get this s.e.s (or complex ?) because all objects appearing in $(\Delta)$ are vector bundles, right ? So by dualizing, we should have something like this : $$0 \to A^* \to H^0(X,A)^* \otimes \mathcal O_X \to E \to 0$$ I suppose that $\omega_C$ appears using the adjunction formula, but I'm lost. Could you help me? Thank you.

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Your $A$ is not a vector bundle on $X$, but a vector bundle on $C$. In particular, it is a torsion sheaf on $X$, so the dual sheaf is zero. But $$ \mathcal{E}\mathit{xt}^1(A,\mathcal{O}_X) \cong A^* \otimes \omega_C, $$ so the second exact sequence comes from the long exact sequence of $\mathcal{E}\mathit{xt}$-sheaves.

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  • $\begingroup$ Thank you a lot, I have no excuses, of course $A$ is not a vector bundle on $X$... everything is clear. Have a nice day. $\endgroup$
    – NaNoS
    Jan 17 at 17:10

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