# Questions tagged [picard-group]

The picard-group tag has no usage guidance.

37
questions with no upvoted or accepted answers

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### 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
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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 $T \mapsto \text{Pic}(X \times_S T)/p^{*}\text{Pic}(T)$, ...

7
votes

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259
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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 ...

6
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### 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
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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 ...

6
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0
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126
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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 $\mathrm{Pic}(A)$...

5
votes

0
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168
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### 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 $\...

5
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0
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469
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### 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 $X$ ...

3
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### 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 $\text{Cl}(X)$ ...

3
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434
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### 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 ...

3
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161
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### 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 ...

3
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138
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### Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?

Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space.
Now let $X'$ ...

3
votes

0
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220
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### How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?

Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...

3
votes

0
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559
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### On generators of the Picard group of a projective smooth surface over a finite field

Let $X$ be a smooth projective surface over a finite field $k=\mathbb{F}_q$. Let us first review the proof of the finite generation of $Pic(X)$ (notice that the proof is valid for any smooth ...

2
votes

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64
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### 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
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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 (...

2
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0
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64
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### 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

0
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83
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### 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 $P$ ...

2
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859
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### 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 $X'$ ...

2
votes

0
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133
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### 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) ...

2
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260
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### Surjectivity of map of Picard schemes implies abelian

Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...

2
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380
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### Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...

2
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### Is Pic( G((z)) ) = $\mathbb{Z}$?

There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In ...

2
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479
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### A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...

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52
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### 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 ...

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0
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161
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### 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 ...

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137
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### 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 $h^{1,1}(X)$?

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0
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### Connection on line bundle over general simplicial toric variety

In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...

1
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0
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47
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### Reference needed for exact sequence of ACM curves with homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence
$$0\to \text{Tor}_i(R,\Bbb{C})_k\to H^1(C,\wedge^{i+1}M_L(...

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0
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200
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### Twisting locally free sheaves in characteristic $p$

Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...

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0
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106
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### Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...

1
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0
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761
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### Picard functor of an algebraic group

Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification ...

1
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### Elegant definition for the scheme parametrizing $g_d^r$'s on a curve

Let $X$ be a smooth projective curve over $k=\bar{k}$, and $Pic^d$ the $d$-part of the Picard group of $X$ (isomorphism classes of line bundles of degree $d$ on $X$).
I'd like to define a scheme $G_d^...

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0
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### $G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-...

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0
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321
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### Abel-Jacobi map for regular fibered surfaces.

Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let $...

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### 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 ...

0
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330
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### Hypersurfaces with Picard group generated by classes of lines on the same plane

For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same ...