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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:
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[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
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