Geodesic
Stable sheave moduli of rank~$2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$
Modelirovanie i analiz informacionnyh sistem, Tome 19 (2012) no. 2, pp. 19-39.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the scheme $M_Q(2;-1,2,0)$ of stable torsion free sheaves of rank $2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on a smooth $3$-dimensional projective quadric $Q$. The manifold $M_Q(-1,2)$ of moduli bundles of rank $2$ with Chern classes $c_1=-1$, $c_2=2$ on $Q$ was studied by Ottaviani and Szurek in 1994. In 2007 the author described the closure $M_Q(-1,2)$ in the scheme $M_Q(2;-1,2,0)$. In this paper we prove that in $M_Q(2;-1,2,0)$ there exists a unique irreducible component different from $\overline{M_Q(-1,2)}$ which is a rational variety of dimension $10$.
Mots-clés : compactification, moduli scheme, $3$-dimensional quadric.
Keywords: coherent torsion free sheave of rank $2$ 
@article{MAIS_2012_19_2_a1,
     author = {A. D. Uvarov},
     title = {Stable sheave moduli of rank~$2$ with {Chern} classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {19--39},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2012_19_2_a1/}
}
TY  - JOUR
AU  - A. D. Uvarov
TI  - Stable sheave moduli of rank~$2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2012
SP  - 19
EP  - 39
VL  - 19
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2012_19_2_a1/
LA  - ru
ID  - MAIS_2012_19_2_a1
ER  - 
%0 Journal Article
%A A. D. Uvarov
%T Stable sheave moduli of rank~$2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$
%J Modelirovanie i analiz informacionnyh sistem
%D 2012
%P 19-39
%V 19
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2012_19_2_a1/
%G ru
%F MAIS_2012_19_2_a1
A. D. Uvarov. Stable sheave moduli of rank~$2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$. Modelirovanie i analiz informacionnyh sistem, Tome 19 (2012) no. 2, pp. 19-39. http://geodesic.mathdoc.fr/item/MAIS_2012_19_2_a1/

[1] D. I. Artamkin, “Svoistva stabilnykh puchkov ranga 2 na trekhmernoi kvadrike”, Yaroslavskii pedagogicheskii vestnik, 1–2, YaGPU, Yaroslavl, 2004, 83–88

[2] P. Khartskhorn, Algebraicheskaya geometriya, Mir, M., 1981 | MR

[3] A. D. Uvarov, “Kompaktifikatsiya mnogoobraziya modulei $M_{Q}(-1,2)$ stabilnykh vektornykh rassloenii ranga 2 na trekhmernoi kvadrike”, Yaroslavskii pedagogicheskii vestnik, 3, Estestvennye nauki, YaGPU, Yaroslavl, 2011, 40–47

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

[5] M. Szurek, J. Wisniewski, “Fano bundles on Fine Moduli Spaces over $\mathbb{P}_{3}$”, Pacific Journal of Mathematics, 141 (1990), 197–208 | MR | Zbl

[6] A. Tyurin, The moduli spaces of vector bundles on threefolds, surfaces and curves I, Erlangen, 1990

[7] J. Wisniewski, “Ruled Fano 4-folds of index 2”, Proceedings of the American Mathematical society, 105, 1983, 55–61 | DOI | MR

[8] J. Ottaviani, M. Szurek, “On moduli of stable 2-bundles with small chern classes on $Q_{3}$”, Annali di Matematica Pura ed Applicata CLXVII, 1994, 191–241 | DOI | MR | Zbl

[9] D. Markushevich, A. Tikhomirov, “The Abel–Jakobi map of a moduli component of vector bundles on the cubic threefold”, J. Algebraic Geometry, 10 (2001), 11–22 | MR

[10] D. Markushevich, A. Tikhomirov, “A parametrization of the theta divisor of quartic double solid”, Intern. Math. Res., 51 (2003), 2747–2778 | DOI | MR | Zbl

[11] N. Perrin, “Mt Limites de fibres vectoriels dans $M_Q_3(0,2)$”, Compte Rendus Acad. Sci., 334, Paris, 2002, 779–782 | MR | Zbl

[12] L. Ein, I. Sols, “Stable vector bundles on quadric hypersurfaces”, Nagoya Math. J., 96 (1984), 11–22 | MR | Zbl

[13] R. Hartshorne, “Stable reflexive sheaves”, Math. Ann., 254 (1980), 121–176 | DOI | MR | Zbl

[14] M. Maruyama, “Moduli of stable sheaves. II”, J. Math. Kyoto Univ., 18 (1978), 557–614 | MR | Zbl

[15] D. I. Artamkin, “Komponenta ne lokalno svobodnykh polustabilnykh puchkov ranga 2 na trekhmernoi kvadrike”, Materialy konferentsii “Chteniya Ushinskogo”, YaGPU, Yaroslavl, 2004, 6–13

[16] C. Banica, M. Putinar, G. Schumacher, “Variation der globalen Ext in Deformationen kompakter komplexer Raume”, Mathematische Annalen, 250 (1980), 135–155 | DOI | MR | Zbl

[17] A. Strømme, “Ample Divisors on fine Moduli spaces on progective plane”, Math. Z., 187 (1984), 405–423 | DOI | MR | Zbl

[18] D. Huybrechts, M. Lehn, “The Geometry of Moduli spaces of sheaves”, Aspects of Mathematics E 31, Vieweg, Braunschweig, 1997 | MR | Zbl

[19] H. Lange, “Universal Families of Extensions”, Journal of Algebra, 83 (1983), 101–112 | DOI | MR | Zbl

[20] K. Hulek, A. Van de Ven, “he Horrocks–Mumford bundle and the Ferrand construction”, Manuscripta Math., 50 (1985), 313–335 | DOI | MR | Zbl