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Stable reflexive sheaves. III. (English) Zbl 0607.14009
The author continues his earlier work [cf. part II of this paper, Invent. Math. 66, 165-190 (1982; Zbl 0519.14008)] on stable reflexive sheaves, by determining, for a rank two stable reflexive sheaf on \({\mathbb{P}}^ 3\) with no global sections (not normalized), the best possible upper bound for the third Chern class \(c_ 3\) as a function of \(c_ 1\) and \(c_ 2.\)
As an application, he finds a bound on the genus \(g\) of a nonsingular curve of degree \(d\) in \({\mathbb{P}}^ 3\) which is not contained in any surface of degree \(<k\). This bound is not the best possible, but it gives strong evidence for the conjectural exact bound also stated in this paper.

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H99 Curves in algebraic geometry
57R20 Characteristic classes and numbers in differential topology
Full Text: DOI EuDML
[1] Ein, L., Hartshorne, R., Vogelaar, H.: Restriction theorems for stable rank 3 vector bundles on ? n . Math. Ann.259, 541-569 (1982) · Zbl 0511.14008
[2] Elencwajg, G., Forster, O.: Bounding cohomology groups of vector bundles on ? n . Math. Ann.246, 251-270 (1980) · Zbl 0432.14011
[3] Gruson, L., Peskine, C.: Postulation des courbes gauches. In: Algebraic geometry (Ravello) Lect. Notes Math. 997, 218-227. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0543.14013
[4] Hartshorne, R.: On the classification of algebraic space curves. In: Vector bundles and differential equations. (Nice). A. Hirschowitz (Ed) 83-112. Basel, Boston, Stuttgart: Birkhäuser 1980
[5] Hartshorne, R.: Stable vector bundles of rank 2 on ?3. Math. Ann.238, 229-280 (1980) · Zbl 0411.14002
[6] Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254, 121-176 (1980) · Zbl 0437.14008
[7] Hartshorne, R., Sols, I.: Stable rank 2 vector bundles on ?3 withc 1=?1,c 2=2. Crelle J.325, 145-152 (1981) · Zbl 0448.14004
[8] Hartshorne, R.: Stable reflexive sheaves. II. Invent. Math.66, 165-190 (1982) · Zbl 0519.14008
[9] Hartshorne, R., Hirschowitz, A.: Cohomology of a general instanton bundle. Ann. Sci. Éc. Norm. Super. IV. Ser.15, 365-390 (1982) · Zbl 0509.14015
[10] Hartshorne, R., Hirschowitz, A.: Nouvelles courbes de bon genre dans l’espace projectif · Zbl 0678.14007
[11] Hartshorne, R.: On the classification of algebraic space curves. II. Proc. AMS Summer Institute in algebraic geometry, Bowdoin (1985) · Zbl 0574.14028
[12] Hirschowitz, A.: Existence de faisceaux réflexifs à bonne cohomologie
[13] Schneider, M.: Chernklassen semistabiler Vektorraumbündel vom Rang 3 auf dem komplex-projektiven Raum. Crelle J.315, 211-220 (1980) · Zbl 0432.14012
[14] Spindler, H.: Der Satz von Grauert-Mülich für beliebige semistabile holomorphe Vektorbündel über demn-dimensionalen komplex-projektiven Raum. Math. Ann.243, 131-141 (1979) · Zbl 0435.32018
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