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by PJPu17
Fri Feb 02, 2018 6:26 pm
Forum: Algebraic Geometry
Topic: Divisors and Picard Group
Replies: 5
Views: 2231

Re: Divisors and Picard Group

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 $Cl(X)=\mathbb{Z}$.
by PJPu17
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 ...
by PJPu17
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}}...
by PJPu17
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...
by PJPu17
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...
by PJPu17
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
by PJPu17
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...
by PJPu17
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...
by PJPu17
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.
by PJPu17
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...