# zbMATH — the first resource for mathematics

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.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14H40 Jacobians, Prym varieties
##### Keywords:
Jacobian; Néron Severi group; Poincaré bundle
Full Text: