# Questions tagged [picard-group]

The picard-group tag has no usage guidance.

123
questions

2
votes

1
answer

175
views

### Does the Grothendieck group detect the Picard group?

Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$.
It is well known that the Grothendieck group $K_0(\operatorname{coh} C)$ of the category of ...

0
votes

0
answers

67
views

### The Anderson dualizing spectrum and $E$-theory

Hopkins-Lurie have shown that the fiber of $gl_1 E \to L_n gl_1 E$ (shown by Ando-Hopkins-Rezk to be coconnected and torsion) looks very very similar to a shift of the Anderson dual of the sphere (in ...

2
votes

0
answers

64
views

### Map to study $K(n)$-local Picard Group

Let $R$ be an $E_{\infty}$-ring. There's a fiber/cofiber sequence $S$: $gl_1 R \to \text{Pic}(R) \to H(\text{Pic}(R))$, where $\text{Pic}(R) =\pi_0 \text{Pic}(R)$ is the Picard group of $R$. Rotating ...

2
votes

0
answers

74
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### Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification

Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better.
In Liu-Lorenzini-Raynaud (...

3
votes

0
answers

62
views

### Detecting non-principal Weil divisors on normal varieties using curves

Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...

5
votes

1
answer

440
views

### Reference request: Generic k3 surface has Picard number 1

I keep running into the statement that "the generic k3 surface has Picard rank 1".
For instance the answer of this question (end) and this paper (following Example 1.1) or this paper (proof ...

1
vote

0
answers

52
views

### Class groups and zeta functions for maximal orders in CSAs

I'm looking for references for certain algebraic objects in the context of maximal orders in (finite dimensional) central simple algebras over algebraic number fields. Does anyone know of any good ...

3
votes

1
answer

250
views

### A question on effective divisors

Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...

2
votes

1
answer

156
views

### Picard group of moduli of principal bundles

I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...

2
votes

2
answers

234
views

### Galois stable elements of the Picard group of a curve and the rational divisors

Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...

3
votes

1
answer

336
views

### Specializing p-torsion in a family of elliptic surfaces

Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$. Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective ...

1
vote

1
answer

152
views

### Picard groups of determinantal varieties

Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...

3
votes

1
answer

138
views

### Picard group of $(SL(n)\times SL(m))$-orbits

Let $\mathbb{P}^N$ be the projective space of $n\times m$ matrices with complex entries modulo scalar. Consider the $(SL(n)\times SL(m))$-action on $\mathbb{P}^N$ given by $((A,B),Z)\mapsto AZB^{T}$. ...

4
votes

1
answer

326
views

### p-torsion in the Picard group of a regular projective curve

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...

2
votes

1
answer

334
views

### Picard group of $\mathrm{GL}(n)$-orbits

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...

9
votes

2
answers

767
views

### 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\...

2
votes

0
answers

64
views

### Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map
$$
Coh(X)\rightarrow Pic(X)\times \mathbf{...

2
votes

1
answer

181
views

### Very general quartic hypersurface in $\mathbb{P}^3$ has Picard group $\mathbb{Z}$

I am looking for a reference from which I can cite the following statement:
The Picard group of a very general quartic hypersurface $X\subset\mathbb{P}^3$ is generated by the class of a hyperplane ...

5
votes

1
answer

537
views

### Picard group of connected linear algebraic group

Here's a statement:
Suppose $G$ is a connected linear algebraic group over a field $k$, then $Pic(G)$ is a finite group.
I know this is true when $k=\mathbb{C}$. My question is does this true for ...

2
votes

1
answer

216
views

### Picard group of a cone over the elliptic curve

Let E be an smooth elliptic curve in a projective plane.
Suppose that X is the projective cone over E in a projective space of dimension three.
What is the Picard group of X ?

7
votes

2
answers

833
views

### 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 $R$ ...

11
votes

1
answer

152
views

### 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 ...

9
votes

1
answer

275
views

### 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 ...

6
votes

0
answers

147
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 ...

1
vote

0
answers

161
views

### Picard scheme of family of quartic surfaces

Recall that a quartic surface in $\mathbb{P}^3_\mathbb{C}$ has $N = 35$ coefficients. Let $U$ be the open subset of $\mathbb{P}^{N-1}$ parametrising smooth quartic surfaces and let $Q \to U$ be the ...

2
votes

0
answers

83
views

### Compactification of Picard variety over normal, projective varieties

Let $X$ be a normal, projective, integral variety (over $\mathbb{C}$) and $P$ be the Picard variety parametrizing invertible sheaves on $X$. Does there exist a compactification $\overline{P}$ of $P$ ...

5
votes

2
answers

431
views

### Picard group of symplectic group modulo orthogonal group

Let $Sp(2n)$ be the group of complex symplectic $2n\times 2n$ matrices, and $O(2n)$ the group of complex orthogonal $2n\times 2n$ matrices.
Consider $Sp(2n)\cap O(2n)\subset Sp(2n)$ and the quotient $...

15
votes

1
answer

632
views

### 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 ...

2
votes

1
answer

240
views

### Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...

3
votes

1
answer

179
views

### Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?

Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...

5
votes

0
answers

168
views

### Picard group of a normal crossing scheme

I would like to know if someone has an explicit example for the rank of the Neron-Severi group of a normal crossing scheme (proper over a field) being different from the rank of the kernel of $\...

9
votes

1
answer

434
views

### 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 ...

2
votes

0
answers

859
views

### Picard group of blowup

Let $Y$ be a nonsingular subvariety of a normal, Cohen-Macaulay variety $X$. Further, let $\pi:X'\to X$ be the blowup of $X$ along $Y$.
Question: Is there a formula for the Picard group of $X'$ ...

1
vote

0
answers

137
views

### Hypersurfaces with maximal Picard rank

Is it true that for any $d \ge 4$, there exists a smooth, degree $d$ surface $X$ in $\mathbb{P}^3$ with maximal Picard rank i.e., Picard rank of $X$ equals $h^{1,1}(X)$?

9
votes

2
answers

543
views

### 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)...

3
votes

0
answers

434
views

### Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ...

6
votes

1
answer

385
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.
...

2
votes

1
answer

655
views

### Extension of line bundle defined over an open subscheme

Let $X$ be a normal projective (or, quasi-projective) variety over $\mathbb{C}$. Let $U \subset X$ be an open subscheme whose complement $Z = X \setminus U$ has codimension at least $2$ in $X$. Let $L$...

2
votes

0
answers

133
views

### The growth of class number in $\mathbb{Z}_p$-extensions of function fields

Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) ...

9
votes

1
answer

508
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 (...

3
votes

1
answer

223
views

### Invertible bimodules which are isomorphic in the stable module category

I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...

9
votes

1
answer

768
views

### 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,\...

4
votes

1
answer

278
views

### Examples of smooth projective varieties with "nice" Picard group

I am looking for examples of smooth projective varieties $(X,H)$ with $H$ a polarization on $X$, $\dim \mbox{Pic}^0(X)=0$, $\mbox{Pic}(X) \not= \mathbb{Z}$ satisfying the property: for any two line ...

8
votes

0
answers

153
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)...

8
votes

2
answers

810
views

### 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) = \...

10
votes

1
answer

359
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 ...

9
votes

2
answers

932
views

### 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)} \...

3
votes

0
answers

161
views

### Explicit algebraic cycles

Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...

5
votes

0
answers

469
views

### Picard group of normalization

Let $X$ be a projective variety with at worst (analytic) normal crossings singularities and $\pi:\tilde{X} \to X$ be the normalisation. Is there a "nice" description relating the picard group of $X$ ...

9
votes

1
answer

462
views

### 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 ...