Picard Functor Research Articles

A Néron model of the universal jacobian

Jacobians of degenerating families of curves are well-understood over 1-dimensional bases due to work of Néron and Raynaud; the fundamental tool is the Néron model and its description via the Picard functor. Over higher-dimensional bases Néron models typically do not exist, but in this paper we construct a universal base change ℳ ˜ g,n →ℳ ¯ g,n after which a Néron model N g,n /ℳ ˜ g,n of the universal jacobian does exist. This yields a new partial compactification of the moduli space of curves, and of the universal jacobian over it. The map ℳ ˜ g,n →ℳ ¯ g,n is separated and relatively representable. The Néron model N g,n /ℳ ˜ g,n is separated and has a group law extending that on the jacobian. We show that Caporaso’s balanced Picard stack acquires a torsor structure after pullback to a certain open substack of ℳ ˜ g,n .

Duality for Commutative Group Stacks

Abstract 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.

Notes on a $p$-adic Exponential Map for the Picard Group

For proper flat schemes over complete discrete valuation rings of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard functor is representable and smooth, our construction recovers and gives finer information to the isomorphism coming from its formal completion. An alternative proof of an old theorem of Mattuck is given.

Classifying affine line bundles on a compact complex space

AbstractThe 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

A stability condition for line bundles on reducible varieties

Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of ‘balanced multidegree’ used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined ‘asymptotically’; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on curves, their behavior in terms of stability asymptotically approaches that of line bundles on curves.Using this definition of stability, we prove that over a one-parameter family of varieties having smooth total space, any line bundle on the generic fiber can be extended to a unique semistable line bundle on the (possibly reducible) special fiber, provided the special fiber is not too complicated in a combinatorial sense.

Jumps and Motivic Invariants of Semiabelian Jacobians

Abstract 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.

Semi-factorial nodal curves and N\xe9ron models of jacobians

Following Pépin, we call a family of curves over a discrete valuation ring semi-factorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the case of nodal curves with split singularities, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up with center the non-regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend the result of Raynaud relating Néron models of smooth curves and Picard functors of their regular models to the case of (possibly singular) curves having a semi-factorial model.

Néron’s pairing and relative algebraic equivalence

Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let XK be a proper smooth and geometrically connected scheme over K. Neron defined a canonical pairing on XK between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When XK is an abelian variety, and if one restricts to those 0-cycles supported on K-rational points, Neron gave an expression of his pairing involving intersection multiplicities on the Neron model A of AK over R. When XK is a curve, Gross and Hriljac gave independently an analogous description of Neron’s pairing, but for arbitrary 0-cycles of degree zero, by means of intersection theory on a proper flat regular R-model X of XK. We show that these intersection computations are valid for an arbitrary scheme XK as above and arbitrary 0-cycles of degree zero, by using a proper flat normal and semifactorial model X of XK over R. When XK=AK is an abelian variety, and X=A¯ is a semifactorial compactification of its Neron model A, these computations can be used to study the relative algebraic equivalence on A¯∕R. We then obtain an interpretation of Grothendieck’s duality for the Neron model A, in terms of the Picard functor of A¯ over R. Finally, we give an explicit description of Grothendieck’s duality pairing when AK is the Jacobian of a curve of index one.

On lifting singular curves

The aim of this note is to lift singular curves of a certain type to characteristic zero. The liftings are obtained as suitable pushouts and the corresponding relative jacobians are identified as rigidifed Picard functors.

Modèles semi-factoriels et modèles de Néron

Let S be the spectrum of a discrete valuation ring with function field K. Let X be a scheme over S. We will say that X is semi-factorial over S if any invertible sheaf on the generic fiber XK can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a proper, flat, normal and semi-factorial model over S. We also construct some semi-factorial compactifications of regular S-schemes, such as Neron models of abelian varieties. The semi-factoriality property for a scheme X/S corresponds to the Neron property of its Picard functor. In particular, one can recover the Neron model of the Picard variety \({{\rm Pic}_{X_K/K,{\rm red}}^0}\) of XK from the Picard functor PicX/S, as in the known case of curves. This provides some information on the relative algebraic equivalence on the S-scheme X.

Finiteness theorems for the Picard objects of an algebraic stack

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaudʼs relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron–Severi groups or of the Picard group itself. We give some examples and applications. In Appendix A, we prove the semicontinuity theorem for a (non-necessarily tame) algebraic stack.

The relative Picard functor on schemes over a symmetric monoidal category

We consider schemes ( X , O X ) over an abelian closed symmetric monoidal category ( C , ⊗ , 1 ) . Our aim is to extend a theorem of Kleiman on the relative Picard functor to schemes over ( C , ⊗ , 1 ) . For this purpose, we also develop some basic theory on quasi-coherent modules on schemes ( X , O X ) over C.

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.

Foncteur de Picard d’un champ algébrique

Resume In this article we study the Picard functor and the Picard stack of an algebraic stack. We give a new and direct proof of the representability of the Picard stack. We prove that it is quasi-separated, and that the connected component of the identity is proper when the fibers of X are geometrically normal. We study some examples of Picard functors of classical stacks. In an appendix, we review the lisse-etale cohomology of abelian sheaves on an algebraic stack.

Semistable degenerations and period spaces for polarized K3 surfaces

Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of logarithmic geometry in the sense of Fontaine and Illusie. The relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces is discussed, and it is explained how the techniques of this paper yield models for the latter spaces over number fields. The paper also contains a discussion of Picard functors for log schemes and a logarithmic version of Artin's method for proving representability by an algebraic stack.

Néron models, Lie algebras, and reduction of curves of genus one

Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.

Groupes vectoriels et schéma de Picard

This Note contains slight variations on a well known theme: linear sub-groups of the Picard functor of a proper scheme over a field k. In particular, we give exemples of a field k, with positive characteristic, and a projective k-scheme X, normal, but not geometrically reduced, such that the neutral component of its Picard functor is representable by a nonzero vectorial group scheme. To cite this article: M. Raynaud, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Rigidified Picard functor and extensions of abelian schemes

Rigidified Picard functor and extensions of abelian schemes

Heights and Arakelov's Intersection Theory

Introduction. This paper relates the Neron-Tate height on Jacobian varieties to an intersection theory on arithmetic surfaces constructed by S. Arakelov [A1 ]. Here I extend and prove a statement of Arakelov to the effect that his intersection pairing is a Neron pairing [A2I. I use this to prove an analogue of the Hodge Index theorem for arithmetic surfaces, a result due independently to G. Faltings. The results of this paper have been extracted from my thesis, written under the direction of Michael Artin. It differs from the thesis in three ways: Firstly it employs the language of local height functions as defined by A. Neron and J. Tate and expounded by S. Lang [LI ]. In the thesis I used Neron's language of local intersection pairings [N1 ]. The second major difference is in the proof of Theorem 1.3. In my thesis I proved this result by using work of Raynaud and Picard Functors [R 1 ], this method was suggested by M. Artin. In this paper I provide a simpler proof observed by S. Lang (and independently) by B. Gross. Finally the development of Arakalov's theory in the context of Neron Functions in section 2 is due to S. Lang, who has kindly given me leave to publish it. This exposition of my results is due in considerable part to S. Lang; it is essentially an edited transcript of a series of conversations between us, which he provided. It is my great pleasure to thank both him and Michael Artin for their help and encouragement. 1. The Neron pairing. Neron [Ne] has given a pairing between divisors and points on abelian varieties, and also on arbitrary varieties, under suitable conditions. We shall give a complement to the Neron theory on curves. Let C be a curve, by which we mean a complete regular curve, geometrically irreducible over a field K with an absolute value v. Let

On the cohomologies of commutative affine group schemes

Let k be an algebraically closed field of positive characteristic p, x a proper integral k-scheme o f finite type and G b e a commutative affine k-group scheme. For a k-prescheme T , the isomorphism classes o f principal fibre spaces Y over X T with group G form an abelian group with the well-known multiplication. We shall denote this abelian group by PH(G, X / k ) ( T ) . Then the functor X / k)(T ) is a contravariant functor from the category o f k-preschemes (Sch/k) to the category o f abelian groups (Ab). The associated sheaf o f PH(G, X / k ) with respect to th e (fpqc)topology of (Sch/k) is denoted by PH(G, X/k). If G is the multiplicative group Gm ., PH(G,„, X/ k ) coincides with the Picard functor Pic(X/k) o f X, and P ic (X / k ) is representable by a commutative k-group scheme, locally of finite type over k. The purpose o f this paper is to study the representability of the functor PH(G, X /k ) for an arbitrary commutative affine k-group scheme of finite type. If G is the additive group G „, PH(G„ X/k) is representable by L ie(P ic(X / k)) which is isomorphic to a direct product of G . If G is a simple finite k-group scheme (i.e. G=trp, P p > (Z/ PZ)k and (Z / qZ ),; q : prime, (p , q )= 1 ), P H (G , X / k ) is * ) This article was presented as a d octoral thesis to the Faculty of Science.