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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: $$(1,1,2),~(2,2,2),~(2,3,2),~(2,3,3),~(2,3,5),~(2,4,r),~2 \le r \le 8.$$
MSC:
14F99Homology and cohomology theory (algebraic geometry)
14J99Surfaces and higher-dimensional varieties
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References:
[1] Ancona V., Ottaviani G.: Some applications of Beilinson’s theorem to projective spaces and quadrics. Forum Math. 3(2), 157--176 (1991) · Zbl 0725.14009 · doi:10.1515/form.1991.3.157
[2] Anghel, C., Manolache, N.: Globally generated vector bundles on $${$\backslash$mathbb{P}\^n}$$ P n with c 1 = 3. arXiv:1202.6261 [math.AG] (2012) (preprint) · Zbl 1279.14053
[3] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer, New York (1985) · Zbl 0559.14017
[4] Arrondo E.: A home-made Hartshorne-Serre correspondence. Rev. Mat. Comput. 20(2), 423--443 (2007) · Zbl 1133.14046
[5] Ballico, E., Huh, S., Malaspina, F.: Globally generated vector bundles of rank 2 on a smooth quadric threefold. J. Pure Appl. Algebra (to appear). arXiv: 1211.1100 [math.AG] (2012) (preprint) · Zbl 1281.14038
[6] Ballico, E., Huh, S., Malaspina, F.: On higher rank globally generated vector bundles over a smooth quadric threefold. arXiv: 1211.2593v2 [math.AG] (2012) (preprint)
[7] Bănică C.: Smooth Reflexive Sheaves. Rev. Roum. Math. Pures et Appl. 36, 571--593 (1991) · Zbl 0773.14005
[8] Chang M.-C.: A filtered Bertini-type theorem. J. Reine Angew. Math. 397, 214--219 (1989) · Zbl 0663.14008
[9] Chiodera L., Ellia P.: Rank two globally generated vector bundles with c 1 5. Rend. Istit. Mat. Univ. Trieste 44, 1--10 (2012) · Zbl 1271.14020
[10] Elencwajg, G., Forster, O.: Bounding cohomology groups of vector bundles on P n . Math. Ann. 246(3), 251--270 (1979/1980) · Zbl 0432.14011
[11] Ellia, P.: Chern classes of rank two globally generated vector bundles on $${$\backslash$mathbb{P}2}$$ P 2 . arXiv:1111.5718 [math.AG] (2011) (preprint)
[12] Hartshorne R.: Algebraic geometry Graduate Texts in Mathematics, vol. 52. Springer, New York (1977) · Zbl 0367.14001
[13] Hartshorne R.: Stable reflexive sheaves. Math. Ann. 254(2), 121--176 (1980) · Zbl 0431.14004 · doi:10.1007/BF01467074
[14] Manolache, N.: Globally generated vector bundles on $${$\backslash$mathbb{P}3}$$ P 3 with c 1 = 3. arXiv:1202.5988 [math.AG] (2012) (preprint)
[15] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston (1980) · Zbl 0438.32016
[16] Sierra J.-C., Ugaglia L.: On globally generated vector bundles on projective spaces. J. Pure Appl. Algebra 213(11), 2141--2146 (2009) · Zbl 1166.14011 · doi:10.1016/j.jpaa.2009.03.012
[17] Sierra, J.C., Ugaglia, L.: On globally generated vector bundles on projective spaces II. J. Pure Appl. Algebra (to appear). arXiv:1203.0185 [math.AG] (2012) (preprint) · Zbl 06244901