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Divisors and Picard Group

Algebraic Geometry
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PJPu17
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Divisors and Picard Group

#1

Post by PJPu17 » Tue Jan 16, 2018 5:10 pm

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 $Spec(\mathbb{C}[x,y,z]/(xy-z^{2})\subset \mathbb{A}^{3}_{\mathbb{C}}$
Tsakanikas Nickos
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Re: Divisors and Picard Group

#2

Post by Tsakanikas Nickos » Thu Jan 18, 2018 1:03 am

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?
PJPu17
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Re: Divisors and Picard Group

#3

Post by PJPu17 » Thu Jan 18, 2018 3:45 pm

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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Re: Divisors and Picard Group

#4

Post by Tsakanikas Nickos » Sun Jan 28, 2018 10:49 pm

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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Re: Divisors and Picard Group

#5

Post by PJPu17 » Fri Feb 02, 2018 6:26 pm

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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Re: Divisors and Picard Group

#6

Post by Tsakanikas Nickos » Wed Mar 07, 2018 10:53 am

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.
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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