Questions tagged [picard-group]
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15 questions
22
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4
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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 ...
- 65.4k
18
votes
1
answer
802
views
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 ...
- 21.4k
5
votes
2
answers
984
views
Injectivity under flat base change of the Picard group on smooth projective curves
Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
- 503
34
votes
4
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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 ...
- 65.4k
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 ...
- 65.4k
11
votes
1
answer
1k
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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
10
votes
1
answer
857
views
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 ...
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2
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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) = \...
- 2,017
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
9
votes
1
answer
607
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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 ...
- 938
9
votes
1
answer
736
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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 (...
- 4,316
7
votes
0
answers
245
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 ...
- 7,377
4
votes
1
answer
194
views
When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
- 167
2
votes
1
answer
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Proposition 1.5 in Mumford's Geometric Invariant Theory
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...
- 5,966
2
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1
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398
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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\...
user168611