# Properties of the Lazarsfeld-Mukai vector bundle - Dualizing an exact sequence of vector bundles

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.

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.
• 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. Jan 17 at 17:10