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Groupe de Picard des variétés de modules de faisceaux semi-stable sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Picard group of the moduli varieties of semi- stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\). (French) Zbl 0616.14006

The subject of this paper is the Picard group of the moduli variety \(M(r,c_ 1,c_ 2)\) of semi-stable algebraic sheaves on \({\mathbb{P}}_ 2({\mathbb{C}})\), of \(rank\quad r\) and Chern classes \(c_ 1, c_ 2\). The first result is that if \(M(r,c_ 1,c_ 2)\) is locally factorial, so \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to the group of linear equivalence classes of Weil divisors of \(M(r,c_ 1,c_ 2)\). There is a unique map \(\delta: {\mathbb{Q}}\to {\mathbb{Q}}\) such that \(\dim (M(r,c_ 1,c_ 2))>0\) if and only if \((c_ 2-(r-1)c^ 2_ 1/2r)/r\geq \delta (c_ 1/r)\). Then if one has equality, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}\), and if the inequality is strict, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}^ 2\). A description of \(Pic(M(r,c_ 1,c_ 2))\) is given, using a subgroup of the Grothendieck group \(K({\mathbb{P}}_ 2)\) of \({\mathbb{P}}_ 2\). It is then possible to compute the canonical bundle \(\omega _ M\) of \(M(r,c_ 1,c_ 2)\).

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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