The Picard scheme

Separated quotients of Picard schemes

We give some necessary and sufficient conditions for the existence of Néron models of jacobians of semistable morphisms of arbitrary relative dimension over base schemes of arbitrary dimension. To do this, we introduce a notion of alignment for semistable morphisms over any regular base scheme, and show that the jacobian of an aligned projective semistable morphism admits a separated model with the Néron mapping property. When the Picard scheme is smooth over the base scheme along its unit section we show that the converse holds.

The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.

The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the associated Abel map A_L : C -> P, which maps C into its compactified Jacobian scheme P, and form its pullback map A_L^* : Pic^0_P -> J, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then A_L^* is known to be an isomorphism. We prove that A_L^* always extends to a map between the natural compactifications, Pic^-_P -> P, and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps…

Notes on a p-adic exponential map for the Picard group

Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring 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 scheme is available and smooth, it recovers the isomorphism coming from its formal completion. A reinterpretation of an old theorem of Mattuck is given.

The units-Picard complex of a reductive group scheme

Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.

若干商概形的皮卡群

Picard Groups of Some Quot Schemes

本文在复数域上亏格大于零的光滑射影曲线上，对局部自由层的商层 Quot 概形，在秩为正且次数充分大时计算了其皮卡群。

Neron models and Compactified Picard schemes over the moduli stack of stable curves

We prove that there exist some stacks, representable over the stack of stable curves, having the following universal property with respect to Néron models of Jacobians. For every one-parameter family of stable curves, with regular total space, the Néron model of the Jacobian of its generic fibre is isomorphic to the base change of the above stacks via the moduli map of the given family. We also obtain a stack compactification of the universal Picard scheme and hence a geometrically meaningful completion of the Néron model of the Jacobian for every family as above.

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.

Picard groups of Hilbert schemes of curves

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the classes of the generators in terms of some ``natural'' divisor classes.

On behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields

Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $δ$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a…

Picard groups for tropical toric schemes

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the s…

An Abel map to the compactified Picard scheme realizes Poincaré duality

For a smooth algebraic curve X over a field, applying H_1 to the Abel map X -> Pic (X/\partial X) to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H_1(X, Z/ n) -> H^1(X/ \partial X, Z/n(1)) = H^1_c(X, Z/n(1)). We show the analogous statement for the Abel map X/\partial X -> Picbar (X/\partial X) to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H_1(X/ \partial X, Z/n) -> H^1(X, Z/n(1)). In particular, H_1 of this Abel map is an isomorphism. In proving this result, we prove some results about Picbar …

Semi-factorial models and Néron models

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 each invertible sheaf on the generic fiber X_K can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a model over S which is proper, flat, normal and semi-factorial. We also construct some semi-factorial compactifications of regular S-schemes, such as Néron models of abelian varieties. Moreover, the semi-factoriality property for a scheme X/S corresponds to the Néron property of its Picar…

扭子层范畴上的Picard群作用

Picard group action on the category of twisted sheaves

本文研究了概形上扭子层的范畴，证明了张量函子自然同构于恒等函子当且仅当该层同构于结构层，并得出Picard群在导出范畴上作用忠实的推论。

Duality of Albanese and Picard 1-motives

We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are t he homological 1-motive of Lichtenbaum and the motivic $H^1$ of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum's conjecture that the …

计算皮卡方案

Computing Picard Schemes

本文提出了一种计算光滑射影簇皮卡方案 torsion 分量的算法，并探讨了其在同调不变量计算中的应用。

Stacks of cyclic covers of projective spaces

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer-Severi schemes of rank 1, which are not necessarily uniform, and give a presentation of the Picard group for substacks corresponding to smooth triple cyclic covers.

The derived Picard group of an affine Azumaya algebra

We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which…

The Geometry of Stable Quotients in Genus One

Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. When the target is projective space and the domain curve has genus 1, these are smooth proper Deligne-Mumford stacks. In this paper we study the associated coarse moduli schemes. We show these schemes are projective, rationally connected and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, and the cones of ample and effective divisors. In certain cases, we also give a closed formula for the Poincaré polynomial.

The relative units-Picard complex and the Brauer group of a product

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under certain hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem in the case of, e.g., (certain types of) ruled varieties over a field of characteristic zero.

微分概形的Galois理论

Galois theory of differential schemes

提出适用于微分概形态射的Galois理论，推广了线性Picard-Vessiot理论与Kolchin强正规理论。

On ordinarity and lifting fibrations on surfaces

We investigate the relation between the ordinarity of a surface and of its Picard scheme in connection with the problem of lifting fibrations of genus g > 1 on surfaces to characteristic zero.

Mordell-Weil Groups of Linear Systems and the Hitchin Fibration

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrictions of line bundles on the variety. As a consequence, we describe the group of sections of the Hitchin fibration for moduli spaces of Higgs bundles on curves.

Determinant bundles for abelian schemes

To a symmetric, relatively ample line bundle on an abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that $p\equiv -1\mod(4)$ and p<2g, where g is the relative dimension of the abelian scheme. Also, we sh…

Line Bundles and Curves on a del Pezzo Order

Orders on surfaces provide a rich source of examples of noncommutative surfaces. Other than some existence results, very little is known about the various moduli spaces that can be associated to them. Even fewer examples have been explicitly computed. In this paper we compute the Picard and Hilbert schemes of an order on the projective plane ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.

Cartwright-Sturmfels Hilbert 概型

Cartwright-Sturmfels Hilbert Schemes

本文证明当 Picard 秩至多为 2 时，Cartwright-Sturmfels Hilbert 概型光滑且不可约，推广了 Fogarty 与 Maclagan-Smith 的经典定理。

On the group of units and the Picard group of a product

Let S be a reduced scheme and let f: X--> S and g: Y-->S be faithfully flat morphisms locally of finite presentation with geometrically connected and geometrically reduced maximal fibers. We discuss the canonical maps G_{m,S}(X)+G_{m,S}(Y)-->G_{m,S}(X x_{S} Y) and Pic X+Pic Y-->Pic(X x_{S} Y) induced by f and g (both sums are direct). Under certain additional conditions on X,Y,S, f and g, we describe the kernel and cokernel of the preceding maps, thereby extending known results when S is the spectrum of a field.

基础代数超几何笔记。Hilbert 和 Picard 超概形

Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes

本文系统地将代数几何的经典概念扩展到超概形几何，首次发展了超概形的 Grothendieck 相对对偶性，构造了更一般的 Hilbert 超概形和 Picard 超概形，并建立了超曲线到主极化阿贝尔超概形的周期映射。

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $Γ(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.

Structures du cube et fibres d'intersection

We define the notion of a hypercube structure on a functor between two strictly commutative Picard categories which generalizes the notion of a cube structure on a $G_m$-torsor over an abelian scheme. We use this notion to define the intersection bundle of $n+1$ line bundles on a relative scheme $X/S$ of relative dimension $n$ and to construct an additive structure on the functor $I_{X/S}:PIC(X/S)^{n+1}\F PIC(S)$. Finally, we study a section of $I_{X/S}(L_1,...,L_{n+1})$ which generalizes the resultant of $n+1$ polynomials in $n$ variables and we interprete some classical formulas with this fo…