Geodesic
On the structure of rigid semistable sheaves on algebraic surfaces
Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 692-700 
B. V. Karpov. On the structure of rigid semistable sheaves on algebraic surfaces. Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 692-700. http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a5/
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     author = {B. V. Karpov},
     title = {On the structure of rigid semistable sheaves on algebraic surfaces},
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     pages = {692--700},
     year = {1998},
     volume = {64},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $S$ be a smooth projective surface, let $K$ be the canonical class of $S$ and let $H$ be an ample divisor such that $H\cdot K<0$. We prove that for any rigid sheaf $F$ ($\operatorname{Ext}^1(F,F)=0$) that is Mumford–Takemoto semistable with respect to $H$ there exists an exceptional set $(E_1,\dots,E_n)$ of sheaves on $S$ such that $F$ can be constructed from $\{E_i\}$ by means of a finite sequence of extensions.

[1] Kuleshov S. A., Orlov D. O., “Isklyuchitelnye puchki na poverkhnostyakh del Petstso”, Izv. RAN. Ser. matem., 58:3 (1994), 59–93

[2] Kuleshov S. A., Isklyuchitelnye i zhestkie puchki na poverkhnostyakh s antikanonicheskim klassom bez bazisnykh komponent, Preprint No 1, Matem. kolledzh NMU, M., 1994

[3] Okonek K., Shneider M., Shpindler Kh., Vektornye rassloeniya na kompleksnykh proektivnykh prostranstvakh, Mir, M., 1984 | Zbl

[4] Mukai S., “On the moduli spaces of bundles on K3 surfaces, I”, Vector Bundles, eds. Atiyah et al., Oxford Univ. Press, 1986, 67–83