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 Post subject: Divisors and Picard GroupPosted: Tue Jan 16, 2018 5:10 pm

Joined: Mon Oct 17, 2016 9:33 pm
Posts: 10
Hi, I´m studying Hartshorne´s book and I´m stuck with the example II. 6.5.2. This example compute the divisor class group of affine quadric cone $Spec(\mathbb{C}[x,y,z]/(xy-z^{2})$. I´m wondering if we take the projective cone $Proj(\mathbb{C}[x_{0},x,y,z]/(xy-z^2))\subset\mathbb{P}^{3}_{\mathbb{C}}$ how can obtain the divisor class group and the Picard group, and, once we´ve obteined the last two groups how can we deduce the divisor class group and the Picard group of $Spec(\mathbb{C}[x,y,z]/(xy-z^{2})\subset \mathbb{A}^{3}_{\mathbb{C}}$

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 Post subject: Re: Divisors and Picard GroupPosted: Thu Jan 18, 2018 1:03 am
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
Hi!

How is your question related to [Hartshorne / II / 6.5.2]? Could you please explain exactly at which point of this particular example you are stuck?

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 Post subject: Re: Divisors and Picard GroupPosted: Thu Jan 18, 2018 3:45 pm

Joined: Mon Oct 17, 2016 9:33 pm
Posts: 10
Hi !

The example shows that de divisor class group of the affine cone is $\mathbb{Z}/2\mathbb{Z}$. My question is how to compute te Picard group of the cone ,and further, if we know the divisor class group /Picard group of an affine variety (over k algebraically closed) what is the relation between these groups and the divisor class group/Picard group of the proyective closure of the variety? In particular, if we know the divisor class group/Picard group of one of them what we can say about the respective groups in the affine (or projective ) variety.

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 Post subject: Re: Divisors and Picard GroupPosted: Sun Jan 28, 2018 10:49 pm
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
Hi!

Let me mention the following, which you may find helpful.

• Recall the following general facts: On a variety $X$, say, over $\mathbb{C}$, it holds that $\text{CaCl}(X) \cong \text{Pic}(X)$. Moreover, if $X$ is normal, then Cartier divisors on $X$ correspond to (are identified with) locally principal Weil divisors on $X$. Finally, if $X$ is locally factorial (in particular if $X$ is smooth), then we have isomorphisms $\text{Cl}(X) \cong \text{CaCl}(X) \cong \text{Pic}(X)$.
• Consider also the following example: On the one hand, since $\mathbb{C}[x_{1}, \dots, x_{n}]$ is a UFD, the (Weil) divisor class group of the affine $n$-space $\mathbb{A}^{n}_{\mathbb{C}}$ over $\mathbb{C}$ is trivial, i.e.
$\text{Cl} \big[ \text{Spec} \big( \mathbb{C}[x_{1}, \dots, x_{n}] \big) \big] = 0.$ On the other hand, the (Weil) divisor class group of the projective $n$-space $\mathbb{P}^{n}_{\mathbb{C}}$ over $\mathbb{C}$ is isomorphic to $\mathbb{Z}$, i.e.
$\text{Cl} \big[ \text{Proj} \big( \mathbb{C}[x_{0}, x_{1}, \dots, x_{n}] \big) \big] = \mathbb{Z}.$ Finally, by the previous comment, we have also determined the corresponding Picard groups.

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 Post subject: Re: Divisors and Picard GroupPosted: Fri Feb 02, 2018 6:26 pm

Joined: Mon Oct 17, 2016 9:33 pm
Posts: 10
I appreciate your response, but after thinking a lot, i can not see which is the Picard group of

$$X=Proj(\mathbb C[x,y,z]/(xy-z^2)]\subset \mathbb{P}^{3}$$

I have just proved that $Cl(X)=\mathbb{Z}$.

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 Post subject: Re: Divisors and Picard GroupPosted: Wed Mar 07, 2018 10:53 am
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
PJPu17 wrote:
I cannot see which is the Picard group of $X=Proj(\mathbb C[x,y,z]/(xy-z^2)]\subset \mathbb{P}^{3}$

I don't know either the answer.

Quote:
I have just proved that $Cl(X)=\mathbb{Z}$.

Maybe you could share your computations.

You could also take a look at [Hartshorne / II / Ex. 6.3], which is related to your initial questions.

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