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Extremal moduli varieties of semistable sheaves over $$\mathbb{P}_ 2(\mathbb{C})$$. (Variétés de modules extrémales de faisceaux semi-stables sur $$\mathbb{P}_ 2(\mathbb{C})$$.) (French) Zbl 0755.14005
The main theorem gives a description of some extremal moduli spaces $$M(r,c_ 1,c_ 2)$$ of semistable sheaves over the complex projective plane of rank $$r$$ and with Chern classes $$c_ 1,c_ 2$$. Such a moduli space is called extremal, if its dimension is positive but the dimension of $$M(r,c_ 1,c_ 2-1)$$ is not greater than zero. Since the author needs some vanishing of cohomology groups, he must assume an additional cohomological property of $$M(r,c_ 1,c_ 2)$$, so that the main result is only established for certain extremal moduli spaces.
The description is given as a good quotient of an open set in a vector space by a certain nonreductive algebraic group. The author’s construction is based on the facts that every exceptional bundle over the complex projective plane is determined by its slope and that one can associate to such a bundle a certain triple of exceptional bundles with nice properties [the author and J. LePotier, Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 193-243 (1985; Zbl 0586.14007)]. The author uses his generalized Beilinson spectral sequence for such triples [cf. Math. Ann. 275, 25-48 (1986; Zbl 0578.14013)] to describe the semi-stable sheaves under consideration as cokernels of well-structured morphisms.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D20 Algebraic moduli problems, moduli of vector bundles 14L24 Geometric invariant theory
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##### References:
 [1] Drezet, J.-M., Le Potier, J.: Fibr?s stables et fibr?s exceptionnels sur ?2. Ann. Sc. Norm. Sup?r18, 193-244 (1985) · Zbl 0586.14007 [2] Drezet, J.-M.: Fibr?s exceptionnels et suite spectrale de Beilinson g?n?ralis?e sur ?2(?). Math. Ann.275, 25-48 (1986) · Zbl 0578.14013 · doi:10.1007/BF01458581 [3] Drezet, J.-M.: Fibr?s exceptionnels et vari?t?s de modules de faisceaux semi-stables sur ?2(?). J. Reine Angew. Math.380, 14-58 (1987) · Zbl 0613.14013 · doi:10.1515/crll.1987.380.14 [4] Drezet, J.-M.: Groupe de Picard des vari?t?s de modules de faisceaux semi-stables sur ?2(?). Ann. Inst. Fourier38, 105-168 (1988) · Zbl 0616.14006 [5] Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math.106, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157 [6] Grothendieck, A.: Sur la classification des fibr?s holomorphes sur la sph?re de Riemann. Am. J. Math.79, 121-138 (1957) · Zbl 0079.17001 · doi:10.2307/2372388 [7] Le Potier, J.: Stabilit? et amplitude sur ?2. Vector bundles and differential equations. (Progr. Math, 7, pp. 145-182. Boston: Birkh?user 1980) [8] Le Potier, J.: Fibr?s stables de rang 2 sur ?2(?). Math. Ann.241, 217-256 (1979) · Zbl 0405.14008 · doi:10.1007/BF01421207 [9] Maruyama, M.: Moduli of stable sheaves. II. J. Math. Kyoto Univ.18, 557-614 (1978) · Zbl 0395.14006 [10] Mumford, D., Fogarty, J.: Geometric invariant theory. (Ergeb. Math. Grenzgeb. Bd.34) Berlin Heidelberg New York: Springer 1982 · Zbl 0504.14008 [11] Newstead, P.E.: Introduction to moduli problems and orbit spaces. TIFR (Lect. Notes Math. vol. 51) Berlin Heidelberg New York: Springer 1978 · Zbl 0411.14003
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