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Reflexive and spanned sheaves on $\mathbb P^3$. (English) Zbl 1304.14023
A reflexive sheaf $\cal F$ on $\bbfP^n, n \ge 3$, is spanned in codimension $2$ if the map $e_{\cal F}:H^0(\cal F) \bigotimes \cal O_X \to \cal F$ is surjective outside of finitely many subvarieties of codimension at least $3$. The paper gives a classification of those reflexive sheaves on $\bbfP^3$ that are simultaneously spanned in codimension $2$ and have small first chern class $c_1$ (i.e. $\le 2$). In order to obtain the classification the paper makes use of the cokernels $\cal F_{k,n,L} $ of the maps: $j: \cal O_{\bbfP^n}(-1) \to \cal O_{\bbfP^n}^{\oplus(k+1)}$, $j$ being defined by $k+1$ linearly independent sections of $\cal O_{\bbfP^n}(1)$ having the linear subspace $L$ as their common zero locus. Such cokernels are proved to be torsion-free sheaves of rank $k$ and singular locus $L$. Then the main result of the paper states that there is an indecomposable non-locally free reflexive sheaf on $\bbfP^3$ of rank $r \ge 2$, spanned in codimension $2$ and having small first chern class $c_1$ if and only if the triple $(c_1,c_2,r)$ is one of the following: $$(1,1,2),~(2,2,2),~(2,3,2),~(2,3,3),~(2,3,5),~(2,4,r),~2 \le r \le 8.$$
14F99Homology and cohomology theory (algebraic geometry)
14J99Surfaces and higher-dimensional varieties
Full Text: DOI
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