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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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