Duality for Commutative Group Stacks

We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X: X\longrightarrow A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $\textrm{Pic}_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give two applications of our Albanese morphism. On the one hand, we give a geometric description of the elementary obstruction and of universal torsors (standard tools in the study of rational varieties over number fields). On the other hand, we give some examples of algebraic stacks that satisfy Grothendieck’s section conjecture.

Classifying affine line bundles on a compact complex space

The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let be a compact complex space with . We introduce the affine Picard functor which assigns to a complex space the set of families of linearly -framed affine line bundles on parameterized by . Our main result states that the functor is representable if and only if the map is constant. If this is the case, the space which represents this functor is a linear space over whose underlying set is , where is a Poincaré line bundle normalized at . The main idea idea of the proof is to compare the representability of to the representability of a functor considered by Bingener related to the deformation theory of -cohomology classes. Our arguments show in particular that, for = 1, the converse of Bingener’s representability criterion holds

Jumps and Motivic Invariants of Semiabelian Jacobians

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

FORMAL GROUPS ARISING FROM FORMAL PUNCTURED RIBBONS

We investigate the Picard functor of a formal punctured ribbon. We prove that under some conditions this functor is representable by a formal group scheme.