Two ways to degenerate the Jacobian are the same

A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties or degenerate moduli spaces of sheaves. Here we relate these two different approaches when the base of the family is a regular, 1-dimensional scheme such as a smooth curve. Specifically, we provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the N\'eron model. The result applies to moduli spaces constructed by Eduardo Esteves and Carlos Simpson, extending results of Busonero, Caporaso, Melo, Oda, Seshadri, and Viviani.Comment: Preprint updated to match published version. Previously appeared as "Degenerating the Jacobian: the N\'eron Model versus Stable Sheaves

The resolution property of algebraic surfaces

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods extend to arbitrary $2$-dimensional schemes that are proper over a noetherian ring.Comment: 19 page

Computing noncommutative deformations of presheaves and sheaves of modules

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures, and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category, and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the last part of the paper, we describe noncommutative deformation functors for sheaves and quasi-coherent sheaves of modules on a ringed space $(X, \mathcal{A})$. We show that for any good $\mathcal{A}$-affine open cover $\mathsf{U}$ of $X$, the forgetful functor $\mathsf{QCoh}(\mathcal{A}) \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces an isomorphism of noncommutative deformation functors. \emph{Applications.} We consider noncommutative deformations of quasi-coherent $\mathcal{A}$-modules on $X$ when $(X, \mathcal{A}) = (X, \mathcal{O}_X)$ is a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of $X$ closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left $\mathcal{D}_X$-module $\mathcal{O}_X$ when $X$ is an elliptic curve as an example.Comment: 22 pages, AMS-LaTeX. Some results from earlier versions have been omitted to focus on the main results in the pape

Coverings in p-adic analytic geometry and log coverings II: Cospecialization of the p'-tempered fundamental group in higher dimensions

This paper constructs cospecialization homomorphisms between the (p') versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction (the tempered fundamental group is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). We studied the question for families of curves in another paper. To construct them, we will start by describing the pro-(p') tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties

Analytic representation theory of Lie groups: General theory and analytic globalizations of Harish--Chandra modules

In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra A(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and A(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.Comment: Main file unchanged. Erratum added at the en

On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic

We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.Comment: arXiv admin note: substantial text overlap with arXiv:1103.262

Homogeneous projective bundles over abelian varieties

We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative algebraic groups; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.Comment: Final version, accepted for publication in Algebra and Number Theory Journal; 37 pages. This is a slightly shortened version of v3: Section 6 has been suppressed as well as the proofs of Propositions 4.1 and 4.2; Section 4 has been relegated to the very en

Derived splinters in positive characteristic

This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities by a result of Kov\'acs. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e., as schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning extending Serre/Kodaira type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove "up to finite cover" analogues in characteristic p of many vanishing theorems one knows in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjectureComment: 22 pages, comments welcome

Noetherian approximation of algebraic spaces and stacks

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.Comment: 39 pages; complete overhaul of paper; generalized results and simplified proofs (no groupoid-calculations); added more applications and appendices with standard results on constructible properties and limits for stacks; generalized Thm C (no finite presentation hypothesis); some minor changes in 2,1-2.8, 8.2, 8.8 and 8.9; final versio

Deformations of elliptic fibre bundles in positive characteristic

We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of F. Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.Comment: Many typos fixed (thanks to the referee). Minor improvements in presentatio