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On rigid-analytic Picard varieties. (English) Zbl 1044.14007
Let \(R\) be a complete discrete valuation ring with field of fractions \(K\), and let \(X\) be a smooth, proper, connected rigid analytic space over \(K\) which possesses a \(K\)-rational point. Under the hypothesis that \(X_K\) has a strict semi-stable formal model over \(R\), the authors construct a rigid analytic variety, \(\text{Pic}_{X/K}\), representing the rigid analytic Picard functor of \(X\), and the Poincaré line bundle on \(\text{Pic}_{X/K} \times X\). They show that the connected component of \(\text{Pic}_{X/K}\) is, after a finite base extension, an extension of a smooth abeloid variety (a smooth proper rigid group variety) by an affine torus. They also show that the rigid Néron-Severi group is finitely generated.

MSC:
14G22 Rigid analytic geometry
14K30 Picard schemes, higher Jacobians
14C22 Picard groups
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