On rationally determined line bundles on a family of projective curves with general moduli (Q1110594)

Fix integers g,r,d; set \(\rho:=\rho (g,r,d):=g-(r+1)(g+r-d)\). Let H be the open subset of \(Hilb({\mathbb{P}}^ r)\) parametrizing smooth, irreducible, nondegenerate curves in \({\mathbb{P}}^ r\) with degree d and genus g. Let \(t:\quad F\to H\) be the universal curve. For every Zariski open subset U of H, set \(R(F_ U)=Co\ker (t^*:Pic(U)\to Pic(t^{- 1}(U))\); \(R(F_ U)\) is the group of rationally determined line bundles over U. Here it is proved (using a theorem of Harer) that if \(\rho\geq 2\), for every \(U\neq \emptyset \quad R(F_ U)\) is generated by the class of the relative canonical line bundle and the class of the hyperplane bundle. It would be very interesting to settle the cases \(\rho =0\) and \(\rho =1\), as well as to solve the corresponding problem for other components with \(\rho <0\).