Torsion-free sheaf on $\Bbb P^2$

algebraic-geometrycoherent-sheavessheaf-cohomology

Let $F$ be a coherent torsion-free sheaf on $\Bbb P^2$ and $L \subset \Bbb P^2$ be a line. Assume that there is an isomorphism $f : F_{|L} \to \mathcal O_L^r$ for some $r \in \Bbb N$.

Questions :

1) Does it implies that there is an open neighborhood $U \supset L$ such that $F_{|U} \cong \mathcal O_U^r$ ? Intuitively being trivialisable is an open condition but I can't formalize it.

2) Why does it implies that $c_1(F) = 0$ ? We need a resolution by vector bundles to compute $c_1(F)$ but I don't see how it helps. Alternatively we can pick a generic section of $F$ and look at its zero locus, again I don't see why this trivialization is useful, $U$ is just slightly bigger than expected i.e it can't contains a closed curve.

Bonus question : what is a good way to describe coherent torsion-free sheaves in terms of vector bundles ?

Best Answer

  1. No, this does not follow. For instance you can take any nontrivial stable vector bundle $E$ on $\mathbb{P}^2$ with $c_1 = 0$. Then for generic line $L$ the restriction $E\vert_L$ is trivial by Grauert-Mulich theorem. On the other hand, if $E\vert_U$ is trivial for some open neighborhood of $L$, then the complement of $U$ is zero-dimensional, hence the isomorphism $E\vert_U \to \mathcal{O}_U^r$ extends (via the pushforward along the inclusion $U \to \mathbb{P}^2$) to an isomorphism $E \to \mathcal{O}_{\mathbb{P}^2}^r$.

  2. Assume $F$ is torsion-free and let $E = F^{\vee\vee}$ be its reflexive hull. Then $E$ is locally free, the natural morphism $F \to E$ is injective, and the cokernel is supported on a zero-dimensional scheme. Thus, we have an exact sequence $$ 0 \to F \to E \to Q \to 0 $$ with $E$ locally free and zero-dimensional $\operatorname{supp}(Q)$. If the line $L$ intersects the support of $Q$, then $Tor_1(\mathcal{O}_L,Q) \ne 0$, hence $F\vert_L$ is not locally free. Thus, the intersection is empty, hence $F\vert_L \cong E\vert_L$. Since $E$ is locally free, we conclude that $c_1(E) = 0$. On the other hand, since the support of $Q$ is zero-dimensional, it does not affect $c_1$, hence $c_1(F) = c_1(E) = 0$.

Bonus question: Any torsion-free sheaf on a smooth surface can be obtained from an exact sequence as above.

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