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Poincaré bundles for families of curves. (English) Zbl 0566.14013
Let X be a curve of genus g over a field k and \(J^ r\) its Jacobian of degree r. Assume that the Néron Severi group of \(J^ r\) is \({\mathbb{Z}}\). We show that there exists a Poincaré bundle on \(J^ r\times X\) if and only if there exists a divisor on X defined over k, of degree \(\delta\) such that \(1-g+r\) and \(\delta\) are coprime. We apply this result to the generic plane curve, the generic curve of a given genus and the generic hyperelliptic curve.

14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H40 Jacobians, Prym varieties
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