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Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques. (Picard groups of moduli varieties of semi- stable bundles on algebraic curves). (French) Zbl 0689.14012
Let X be a smooth projective curve of genus \(g\geq 2\) over \({\mathbb{C}}\). Let U(r,d) (resp. \(U_ s(r,d))\) be the moduli space of algebraic semistable vector bundles (resp. the open subset corresponding to the stable bundles) of rank \(r\geq 2\) and degree d over X. It is known that \(U(r,d)\) is a normal, irreducible, projective variety. If \(gcd(g,r)\neq 1\) and one excludes also the case \(g=r=2\), d even then \(U(r,d)\) is not smooth, \(Sing(U(r,d))=U(r,d)\setminus U_ s(r,d)\) and \(co\dim_{U(r,d)}U(r,d)\setminus U_ s(r,d)\geq 2\). For \(L\in Pic(X)\), \(\deg (L)=d\) let denote by U(r,L) (resp. \(U_ s(r,L))\) the closed subvariety of \(U(r,d)\) (resp. \(U_ s(r,d))\) corresponding to the vector bundles with determinant isomorphic to L. The aim of this paper is to give a complete description of \(Pic(U(r,d))\) and \(Pic(U(r,L))\) when \(gcd(g,r)\neq 1\) and \((g,r)\neq (2,2)\), d even.
The first result is that even they are singular, \(U(r,d)\) and \(U(r,L)\) are locally factorial. Let now \(\gcd (r,d)=n\) and let \({\mathcal F}\) be a vector bundle on X such that \(\deg({\mathcal F})=(-d+r(g-1))/n\) and \(rk({\mathcal F})=r/n\). Then \(\chi({\mathcal E}\otimes {\mathcal F})=0\) for all vector bundles \({\mathcal E}\) on X of rank r and degree d. One can show that \({\mathcal F}\) above can be chosen such that there exists \({\mathcal E}\in U_ s(r,d)\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})=H^ 1(X,{\mathcal E}\otimes {\mathcal F})=0\). Then for such an \({\mathcal F}\) denote by \(\Theta^ s_{{\mathcal F}}\) (respectively \(\Theta^ s_{{\mathcal F},L})\) the set of points of \(U_ s(r,d)\) (resp. \(U_ s(r,L))\) which correspond to stable bundles \({\mathcal E}\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})\neq 0\). These are showed to be hypersurfaces in \(U_ s(r,d)\) respectively in \(U_ s(r,L)\). Their closure in \(U(r,d)\) (respectively \(U(r,L)\)) are denoted by \(\Theta_{{\mathcal F}}\) (resp. \(\Theta_{{\mathcal F},L})\) and called theta divisors.
The line bundle \({\mathcal O}(\Theta_{{\mathcal F},L})\) is independent of the choice of \({\mathcal F}\) and \(Pic(U(r,L))\) is isomorphic to \({\mathbb{Z}}\) having \({\mathcal O}(\Theta_{{\mathcal F},L})\) as generator. Let \(I^{(d)}\) be the Jacobian of the line bundles of degree d on X. Then, through the canonical morphism \(\det: U(r,d)\to I^{(d)},\) \(Pic(I^{(d)})\) is seen as a subgroup of Pic(U(r,d)) and one has the isomorphism \(Pic(U(r,d))\cong Pic(I^{(d)})\oplus {\mathbb{Z}}{\mathcal O}(\Theta_{{\mathcal F}})\). Here \({\mathcal O}(\Theta_{{\mathcal F}})\) is dependent on the choice of \({\mathcal F}:\) \({\mathcal O}(\Theta_{{\mathcal F}'})\cong {\mathcal O}(\Theta_{{\mathcal F}})\otimes \det^*(\det {\mathcal F}'\otimes (\det {\mathcal F})^{-1}).\)
The paper also contains a complete description of the dualizing sheaves of \(U(r,L)\) and \(U(r,d)\) and a proof of the nonexistence of Poincaré bundles on open subsets of the moduli space \(M_ s({\mathbb{P}}_ 2({\mathbb{C}}),r,c_ 1,c_ 2)\) in case r, \(c_ 1\) and \(\chi\) are not prime to each other.
Reviewer: Sorin Popescu

MSC:
14C22 Picard groups
14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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