Mathematics

Questions tagged [picard-group]

19 questions
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Example of a Picard stack

Let $F$ be the stack (over some base scheme $S$ ) which associates to every scheme $X$ the groupoid of invertible sheaves on $X$ . Then by the general theory (see for example Aoki's paper ) $F$ is algebraic. My question is: Can you write down a nice…
Martin Brandenburg
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Why is the sequence $0\rightarrow\mathscr{O}_C(C)^\ast\rightarrow k(C)^\ast\rightarrow\rm Div(C) \rightarrow Pic(C)\rightarrow 0$ exact?

Definition A curve over a field $k$ is a separated scheme $C$ of finite type over $k$ which is integral of dimension 1. Let $C$ be a normal curve over a field $k$ . A divisor is an element of the free abelian group…
Ziqiang Cui
  • 553
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What is the Picard group of orthogonal grassmanians?

I am interested in understanding the Picard group of orthogonal Grassmanians in the following two cases. If someone could give me the proofs or a hint or a good reference it would make me happy. Case 1: Let $V$ be an even dimensional vector space…
frogorian-chant
  • 914
4
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1 answer

Give an example of a commutative ring with unity $R$, such that $K_0(R) \ncong K_0(R[t])$

In Weibel's $K$ -book I have read that it is Quillen's classical result that if $R$ is a regular Noetherian ring then $K_0(R) \cong K_0(R[t])$ . So out of curiosity I have tried and failed quite a lot of times to see what happens for non-regular…
Divya
  • 603
4
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$R$-modules satisfying $M^{\otimes n}\simeq R$ but not $M^n\simeq R^n$

This question is a byproduct of the following question . Let $R$ be a commutative ring with unit. Using exterior algebra computations, one may show that if an $R$ -module $M$ satisfies $M^n\simeq R^n$ , then $M$ is a projective module of rank one…
GreginGre
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3
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A line bundle $L$ can be assumed to be ample (after eventually passing to its dual)

I'm reading Huybrechts' Lectures on K3 Surfaces and I got stuck reading example 2.3.9, which shows that any K3 surface $X$ with $\operatorname{Pic}(X)=\mathbb{Z}\cdot L$ and such that $(L)^2=4$ can be realized and a quartic $X \subset \mathbb{P}^3$…
WindUpBird
  • 417
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reference and full statement for the fundamental group of $\Bbb{P}_n^k-Y$ is $\Bbb{Z}/(d)$?

In Vakil’s FOAG, exercise 15.4.M, there is a remark: 15.4.M. EXERCISE: A TORSION PICARD GROUP. Suppose that $Y$ is a hypersurface in $\Bbb{P}^n_k$ corresponding to an irreducible degree d polynomial. Show that $Pic(\Bbb{P}^n_k-Y) = Z/(d)$ . (For…
onRiv
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Automorphisms of matrix algebras and Picard Group

Notation. In what follows, $R$ is a commutative ring with $1$ , $n\geq 1$ is an integer, $\mathcal{B}=(e_1,\ldots,e_n)$ is a fixed basis of $R^n$ , and $e_{ij}$ is the usual elementary endomorphism, namely the unique endomorphism of $R^n$ such that…
GreginGre
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Hartshorne Algebraic Geometry Exercise III.11.5 (On Picard groups of Formal Completions)

Let $\widehat{X}$ be the formal completion of $X=\mathbb{P}^N_k$ along a hypersurface for $N\geq 4$ . The exercise is to prove $\operatorname{Pic}(\widehat{X})\rightarrow \operatorname{Pic}(Y)$ is an isomorphism. The hint is to use Exercise II.9.6,…
user992440
2
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Exactness of a sequence arising from the Hochschild-Serre sequence in étale cohomology

Disclaimer: I'm not an expert in étale cohomology. This just came up, when trying to generalize a method, that came up studying a problem in algebraic number theoy/ class field theoy. Let $f:Y\to X$ be a Galois cover of schemes. Then for an étale…
Firebolt2222
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Tools for computing Picard groups

I have a particular graded ring whose Picard group I am trying to compute, namely $$\mathbb{Z}[x,y,z]/(2x, x^3, xy, z^2-4y),$$ where $|x|=1$ , $|y|=4$ , and $|z|=8$ . How does one go about computing this? I know that if I can do some algebra to…
categorically_stupid
  • 419
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Help understanding the Picard group of a relatively minimal conic bundle over the projective line?

Let $X$ be a projective, geometrically connected $k$ -surface with a relatively minimal conic bundle structure $X \longrightarrow \mathbb{P}^1_k$ . My understanding is that the generic fiber ought to be a smooth, genus zero curve and say we have n…
Sunbeam
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Calculation of the Picard group and the class group

I am thinking about the computation of the class group and the Picard group for the case of Number fields over $\mathbb{Q}$ and $\mathbb{F}_p(t)$ Complex varieties I would like to know what kinds of algorithms exist for the computation of these two…
user900250
2
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1 answer

Conductor ideals and Picard group

Let $k$ be a field. Then for the extension $R = k[t^2, t^3, s] \rightarrow S = k[t, s]$ , I want to figure out the conductor ideal $ I $ of $R $ which is defined to be the largest ideal of S contained in R which is $ I = \{x\in R : xS \subset R\}$…
Divya
  • 603
1
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Show that the subgroup $\{D \in \text{Pic}F: 2D \sim 0\}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$

Let $F = y^2z - x^3 - \lambda xz^2 - \mu z^3$ be a smooth elliptic curve. Show that the subgroup $S = \{D \in \text{Pic}F: 2D \sim 0\}$ of $\text{Pic}F$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ . Let $\Phi$ be the…
ellenying
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