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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## Search found 10 matches

- Fri Feb 02, 2018 6:26 pm
- Forum: Algebraic Geometry
- Topic: Divisors and Picard Group
- Replies:
**5** - Views:
**2231**

- Thu Jan 18, 2018 3:45 pm
- Forum: Algebraic Geometry
- Topic: Divisors and Picard Group
- Replies:
**5** - Views:
**2231**

### Re: Divisors and Picard Group

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 ...

- Tue Jan 16, 2018 5:10 pm
- Forum: Algebraic Geometry
- Topic: Divisors and Picard Group
- Replies:
**5** - Views:
**2231**

### Divisors and Picard Group

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}}...

- Sun Dec 17, 2017 12:58 am
- Forum: Algebraic Geometry
- Topic: Locally free sheaves
- Replies:
**1** - Views:
**821**

### Locally free sheaves

Hi I´m stuck with the following proposition, could you help me to solve it? Suppose $0\rightarrow{\cal{E}}'\rightarrow{\cal{E}}\rightarrow{\cal{E}}''\rightarrow0$ is an exact sequence of locally free sheaves of ranks $r'\,, \, r$ and $r''$. Then \[\Lambda^{r}{\cal{E}}\cong\Lambda^{r'}{\cal{E}}'\otim...

- Sat Oct 07, 2017 3:25 pm
- Forum: Algebra
- Topic: Locally free but no globally
- Replies:
**1** - Views:
**1077**

### Locally free but no globally

Let $R=k[x,y]/(x^{2}+y^{2}-1)$, and let $\mu=(x,y-1)\subset R$. I want to prove that $\mu$ is locally free (i.e that the localization in the multiplicative system defined by each prime $\mathfrak{p}$ is a free $R_{\mathfrak{p}}$-module). I have just proved that $\mu$ is locally free of rank 1, but I...

- Mon Oct 24, 2016 11:04 pm
- Forum: Differential Geometry
- Topic: Parallel
- Replies:
**5** - Views:
**2416**

### Re: Parallel

I can´t do this, because i have the metric induce by the euclidean ( the first fundamental form on the sphere), and these fields are in cartesians. :S

- Mon Oct 24, 2016 1:36 pm
- Forum: Differential Geometry
- Topic: Parallel
- Replies:
**5** - Views:
**2416**

### Re: Parallel

I came up to these point but I don´t know how derivate these fields because they´re in cartesians. On the other hand, Is there any geometric argument to prove the statement without doing any operation? Using only the compatibility of the connection with the metric, and the orthogonality of both fiel...

- Mon Oct 24, 2016 12:20 am
- Forum: Differential Geometry
- Topic: Parallel
- Replies:
**5** - Views:
**2416**

### Parallel

Hi, i´ve working hard on this problem but i don´t get the solution. It is the exercise 2.12 of this notes http://www.maths.ed.ac.uk/~aar/papers/dupontnotes.pdf" onclick="window.open(this.href);return false; I´ve computed Christoffel´ symbols of the induced conection, and the metric´s matrix, but i d...

- Tue Oct 18, 2016 1:52 pm
- Forum: Algebraic Geometry
- Topic: Localization
- Replies:
**3** - Views:
**1462**

### Re: Localization

I understand you but i´m asking for a particular example of topological space where this presheaf is not a sheaf, or an explanation about why this presheaf is not always a sheaf.

- Mon Oct 17, 2016 9:51 pm
- Forum: Algebraic Geometry
- Topic: Localization
- Replies:
**3** - Views:
**1462**

### Localization

Let $R$ a ring , and $ X =Spec(R)$, we define the presheaf of localization by open subsets $ U\subset X$, as $R′(U)=R_{SU}$ wbere $SU=\{ f\in R : (f)_{0}\cap U=\emptyset \}$, Is in a general case a sheaf or do exists some examples of rings where this prehseaf is not a sheaf?, When we work with basic...