# Singularities of reflexive sheaves

I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (on $\mathbb{P}^3$, for higher dimensional varieties it fail to be locally free in a scheme of codimension 3). Therefore, if one restricts a reflexive sheaf in an open of $\mathbb{P}^3$ not containing the singular points, the restricted sheaf should isomorphic to copies of the trivial. I would like to understand what happens when one restrict the sheaf to an open containing the singular point. Is there a way of computing explicit what the restricted sheaf will look like?

• What do you mean by explicit? It won't be more explicit that the description of the sheaf on the whole $\Bbb{P}^3$. – abx Mar 26 '18 at 5:15
• A reflexive sheaf is the kernel of a morphism of locally free sheaves. Locally, you can assume the locally free sheaves to be trivial, so you can think of a reflexive sheaf as a subsheaf of $\mathcal{O}^{\oplus n}$ given by a finite number of fiberwise linear conditions. – Sasha Mar 26 '18 at 6:26
• Dear @abx by explicit I mean explicit in that open, not in the whole $\mathbb{P}^3$, e.g., a not trivial locally free sheaf is not sum of the trivial on whole $\mathbb{P}^3$, but it is in some open subset. – User43029 Mar 26 '18 at 11:43