The Picard group of the universal Picard varieties over the moduli space of curves (Q1189297)

We denote by \({\mathcal M}^ 0_ g\) the moduli space of smooth curves of genus \(g\) \((g\geq 3)\) without automorphisms, and by \({\mathcal C}_ g@>\pi>>{\mathcal M}^ 0_ g\) the universal curve over \({\mathcal M}^ 0_ g\). For any integer \(d\), we denote by \(\psi_ d:{\mathcal T}^ d_ g\to{\mathcal M}^ 0_ g\) the universal Picard (Jacobian) variety of degree \(d\); the fiber \(J^ d(C)\) over a point \([C]\) of \({\mathcal M}^ 0_ g\) parametrizes line bundles on \(C\) of degree \(d\), modulo isomorphism. We describe a group \({\mathcal N}({\mathcal T}^ d_ g)\) (which we call the relative Néron-Severi group of \({\mathcal T}^ d_ g)\) defined to be the group of line bundles on \({\mathcal T}^ d_ g\), modulo the relation that two line bundles are equivalent if their restrictions to the fibers of the map \(\psi_ d\) are algebraically equivalent. Lemma: The Néron-Severi group of the Jacobian of a curve \(C\) with general moduli is generated by the class \(\theta\) of its theta divisor. We can define an embedding of groups \(\varphi_ d:{\mathcal N}({\mathcal T}^ d_ g)\hookrightarrow\mathbb{Z}\). To describe the group \({\mathcal N}({\mathcal T}^ d_ g)\) is equivalent to finding the generator \(k^ d_ g\) of the image of the map \(\varphi_ d\). Theorem: For \(d=0,\ldots,g-1\) the numbers \(k^ d_ g\) are given by the following formula: \(k^ d_ g=(2g-2)/\text{g.c.d}(2g-2,g+d-1)\).