Corrigendum: The affine part of the Picard scheme

We give a corrected version of Theorem 3, Lemma 4, and Proposition 9 in the above-mentioned paper, which are incorrect as stated (as was pointed out by O. Gabber).

The affine part of the Picard scheme

We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.

THE SCHEME OF FORMAL CURVES ON AN ABELIAN VARIETY

Let X be an abelian variety defined over an algebraically closed field k of arbitrary characteristic. The aim of this note is to show that the scheme [Formula: see text], which parametrizes germs of formal curves on X, is the Picard scheme of a formal scheme.

THE STRONG FRANCHETTA CONJECTURE IN ARBITRARY CHARACTERISTICS

Using Moriwaki's calculation of the ℚ-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g ≥ 3. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.