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## Search found 284 matches

Tue Jul 31, 2018 11:51 pm
Forum: Complex Analysis
Topic: Best book(s) for Complex Analysis (undergrad)
Replies: 1
Views: 411

### Re: Best book(s) for Complex Analysis (undergrad)

I would suggest that you read the following books on Complex Analysis: J. Marsden, M. Hoffman - Basic Complex Analysis : Although it is developed in a rather slow manner, the exposition is in my opinion very nice, and the book contains lots of examples as well as exercises that help one acquire a go...
Wed Mar 07, 2018 10:53 am
Forum: Algebraic Geometry
Topic: Divisors and Picard Group
Replies: 5
Views: 1724

### Re: Divisors and Picard Group

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 you...
Sun Jan 28, 2018 10:49 pm
Forum: Algebraic Geometry
Topic: Divisors and Picard Group
Replies: 5
Views: 1724

### Re: Divisors and Picard Group

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 w...
Thu Jan 18, 2018 1:03 am
Forum: Algebraic Geometry
Topic: Divisors and Picard Group
Replies: 5
Views: 1724

### Re: Divisors and Picard Group

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?
Tue Dec 19, 2017 12:40 am
Forum: Algebraic Geometry
Topic: Locally free sheaves
Replies: 1
Views: 679

### Re: Locally free sheaves

Hi!

You can find an answer to your question in the following reference: [Q. Liu - Algebraic Geometry and Arithmetic Curves - Chapter 6 / Lemma 4.1 & Corollary 4.2]
Sun Nov 12, 2017 1:01 am
Forum: Algebraic Geometry
Topic: Geometric Genus
Replies: 1
Views: 806

### Re: Geometric Genus

Let $n = \dim X$. As $X$ is rational, (by definition) $X$ is birationally equivalent to $\mathbb{P}^{n}$, and since the geometric genus $p_{g}(X) (= P_{1}(X) = \dim H^{0} (X, \omega_{X} ) )$ is a birational invariant, we have that $p_{g} (X) = p_{g} (\mathbb{P}^{n})$. But the canonical s...
Sun Mar 05, 2017 12:49 am
Forum: Complex Analysis
Topic: Exercise On Cohomology of Complex Spaces
Replies: 0
Views: 575

Assuming the following result THEOREM : Let $X$ be a complex space of dimension $n$ and let $\mathcal{S}$ be any sheaf on $X$. Then $\mathrm{H}^{q}(X, \mathcal{S}) = 0 \, , \, q > 2n$ prove the following results LEMMA : Let $X$ be a complex space of dimension $n$ such that $\mathrm{H}^{q}(X, \... Sat Mar 04, 2017 10:16 pm Forum: Functional Analysis Topic: An exercise on Fréchet Spaces Replies: 1 Views: 755 ### An exercise on Fréchet Spaces Let V,W be Fréchet spaces and let T be a Hausdorff space. Consider the diagram \[ V \overset{f}{\longrightarrow} W \overset{i}{\longrightarrow} T$
where $i$ is a continuous, linear, injective map and $f$ is a linear map. Show that $f$ is continuous if and only if $i \circ f$ is continuous.
Thu Feb 23, 2017 2:02 am
Forum: Algebraic Structures
Topic: Isomorphism
Replies: 2
Views: 872