Questions tagged [picard-group]
The picard-group tag has no usage guidance.
139
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What is the right definition of the Picard group of a commutative ring?
This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
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votes
2
answers
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Torsion-freeness of Picard group
Let $X$ be a complex normal projective variety.
Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?
One technique I sometimes use is following:
If $X$ can be ...
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21
votes
4
answers
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Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
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votes
1
answer
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Does every relative curve have a Picard scheme?
More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...
- 23.5k
18
votes
1
answer
736
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Relative Picard functor for the Zariski topology
I'm trying to understand better the relative Picard functor, as defined, for example, in Kleiman's article.
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are ...
- 20.6k
17
votes
1
answer
985
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Is the ring of all cyclotomic integers a Bezout domain?
My previous question about the theorem (apparently due to Dedekind -- thanks, Arturo Magidin!) that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is a Bezout domain got me thinking about ...
- 64.4k
15
votes
6
answers
2k
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Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
- 64.4k
15
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1
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731
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Postnikov invariants of the Brauer 3-group
Given a commutative ring $k$ there is a bicategory with
algebras over $k$ as objects,
bimodules as morphisms,
bimodule homomorphisms as 2-morphisms.
This is a monoidal bicategory, since we can ...
- 21k
14
votes
2
answers
2k
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Picard Groups of Moduli Problems
First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is ...
- 15.7k
14
votes
2
answers
3k
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Picard group of a singular projective curve
Let $X$ be a singular irreducible projective curve over an algebraically closed field and $\pi : \widetilde{X} \to X$ the normalization morphism. In the book on Neron models by Bosch et al. (I have ...
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12
votes
2
answers
996
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Why are ordinary spheres not strictly invertible?
Introduction
This question is about Picard spectra for the symmetric monoidal $\infty$-category of spectra. We say that a spectrum $X$ is invertible if there is another spectrum $Y$ such that $X\...
- 54.6k
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1
answer
2k
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Picard groups of non-projective varieties
As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:
If $X$ is proper then $Pic_{X/k}$ is ...
- 4,400
11
votes
1
answer
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Restriction of the Picard group of a surface to a curve
In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:
For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
- 111
11
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1
answer
195
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Are algebras with invertible linear duals always Frobenius?
Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an invertible ...
- 26.9k
10
votes
2
answers
1k
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Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
user168611
10
votes
1
answer
425
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Class numbers of functions fields and spanning trees
In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-...
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10
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1
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406
views
Why is the theorem of the base mostly cited only for smooth proper varieties
This is a very soft question, and I'm not sure what I expect as an answer.
In SGA6, Expose XIII, Theoreme 5.1 it is proven that, if $X$ is a proper scheme over a field $k$, then $NS(X)$ is finitely ...
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1
answer
777
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Picard group generated by effective divisors: counterexample?
Let $X$ be an integral variety defined over an algebraically closed field $k$ of characteristic 0 with finitely generated Picard group $Pic(X)$ and such that $k[X]^\times=k^\times$ (i.e. the only ...
- 101
10
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1
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Picard group of Drinfeld upper half space
Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).
Is the Picard group of $\Omega^{(n)}_K$ known? ...
- 10.5k
9
votes
2
answers
996
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Picard group of a finite type $\mathbb{Z}$-algebra
Let $A$ be a finitely generated $\mathbb{Z}$-algebra. Is $\operatorname{Pic}(A)$ finitely generated (as an abelian group)?
Thoughts:
We may assume that $A$ is reduced since $\operatorname{Pic}(A) = \...
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9
votes
2
answers
634
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On a morphism from the Brauer group to the Picard group
Suppose that $k$ is a commutative ring and that $A$ is an Azumaya
$k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...
- 50.6k
9
votes
1
answer
485
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Pic^0 and H^0(K,Pic^0)
Let $K$ be a field and $C$ a smooth and projective curve over $K$. Then the kernel $Pic^0(C)$ of the degree map injects into $H^0(K,Pic^0_C)$, where $Pic_C^0$ is the connected component of the Picard ...
- 1,413
9
votes
2
answers
1k
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Galois invariant Picard group elements
Let $X$ be a smooth variety over a perfect field $k$ with $X(k) \neq \emptyset$. Then is the natural map
\begin{equation}
\mathrm{Pic}(X) \to (\mathrm{Pic}(X_{\bar{k}}))^{\mathrm{Gal}(\bar{k}/k)} \...
- 20.6k
9
votes
1
answer
645
views
Del Pezzo surfaces and Picard-Lefschetz theory
Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
- 4,246
9
votes
2
answers
819
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$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
- 3,697
9
votes
1
answer
327
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Picard-surjectivity and Morita-equivalence
Let us say that an algebra $A$ over a field $k$ is Picard-surjective if the canonical map
$$ \mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$
is surjective. Here $\mathrm{Pic}(A)$ denotes the group of ...
- 9,481
9
votes
1
answer
916
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Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 3,364
9
votes
1
answer
487
views
Does a semistable curve descend to a regular base?
Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that
$f$ is proper, flat, and of finite presentation;
The ...
- 1,103
9
votes
1
answer
514
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Bézout ring with non-trivial Picard group?
[I asked this on stackexchange here a few weeks ago to no response]
A ring is called Bézout when its finitely generated ideals are principal.
Q: Is there a nice example of a Bézout ring $R$ with ...
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0
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Definition of relative Picard functor
Let $X \to S$ be a morphism of schemes. The relative Picard functor from schemes over $S$ to abelian groups is usually defined by the formula $T \mapsto \text{Pic}(X \times_S T)/p^{*}\text{Pic}(T)$, ...
- 3,505
8
votes
6
answers
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Picard group, Fundamental group, and deformation
One of the most elementary theorems about Picard group is probably $\mathrm{Pic} (X \times \mathbb{A}^n) \cong \mathrm{Pic} X$ and $\mathrm{Pic} (X \times \mathbb{P}^n) \cong \mathrm{Pic} X \times \...
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8
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2
answers
754
views
Are there varieties with non finitely generated Picard group and vanishing irregularity?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...
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votes
1
answer
2k
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
- 1,105
8
votes
1
answer
251
views
Obstructions to Picard-graded groups of maps
Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For $\...
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8
votes
0
answers
165
views
On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?
Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...
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votes
1
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Picard group of $\mathcal{M}_{0,n}$
Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?
- 3,697
7
votes
1
answer
442
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Galois invariant line bundles on a product of varieties
Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and ...
7
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2
answers
2k
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Picard group vs class group
The question.
Let $R$ be a commutative ring. Let $M$ be an $R$-module with the property that there exists an $R$-module $N$ such that $M\otimes_R N\cong R$. Does there always exist an ideal $I$ of $R$ ...
- 40.3k
7
votes
0
answers
204
views
Albanese morphism induces an isomorphism on global $1$-forms
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
- 6,952
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0
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278
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Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
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votes
2
answers
387
views
adjacency matrix of a graph and lines on quartic surfaces
Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in ...
6
votes
1
answer
1k
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What does the Riemann-Hurwitz formula tell us on the Picard variety
Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.
Then we have a linear equivalence of Weil divisors on $X$: $$ K_X=f^\...
- 9,377
6
votes
1
answer
357
views
Picard group of derived category of sheaves
Let $X$ be a topological space and $R$ be a commutative ring with unit, $D(X,R)$ is the derived category of unbounded complexes of sheaves of $R$-modules. Moreover we suppose that $X$ is a stratified ...
- 9,762
6
votes
1
answer
800
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Are Picard stacks group objects in the category of algebraic stacks
I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack.
I'm slightly confused by the terminology here.
Given an algebraic stack $\mathcal X$ ...
- 193
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1
answer
451
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An example where $Pic(X) = H^0(k,Pic(\overline{X}))$?
Let $X$ be a geometrically integral smooth projective variety over a number field $k$. Then if $X$ is everywhere locally soluble, we have $Pic(X) = H^0(k,Pic (\overline{X}))$, where $\overline{X}=X \...
- 20.6k
6
votes
1
answer
420
views
Proper scheme such that every vector bundle is trivial
It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.
...
user138661
6
votes
1
answer
398
views
Severi's theorem of base and Hilbert polynomial
Let $X$ be a smooth projective variety over $\mathbb{C}$ satisfying $H^1(\mathcal{O}_X)=0$. Fix $i:X \to \mathbb{P}^n$ a closed immersion and let $\mathcal{O}_X(1)$ be the corresponding very ample ...
- 2,106
6
votes
1
answer
865
views
Picard groups and birational morphisms
Let $f:X\rightarrow Y$ be a birational morphism of projective varieties. Assume that $Pic(X)$ is a free abelian group generated by $n$ divisors $D_1,...,D_n$.
Under which hypothesis on $X$ and $Y$ is ...
user97096
6
votes
1
answer
791
views
Relatively numerically trivial divisor
Hi,
Let $f : X \rightarrow Y$ a projective morphism of quasi-projective algebraic varieties over $\mathbb{C}$. Assume that $X$ is smooth, that $Y$ is normal and that:
$$\textbf{R} f_* \mathcal{O}_X ...
- 213
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0
answers
129
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Center Picard group non-commutative algebra
I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
- 7,100