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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 ...

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 ...

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 ...

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?

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]

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 ...

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 ...

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.

Consider the composite map
\[ R \overset{\varphi}{\to} R \overset{\pi}{\twoheadrightarrow} R/J \]
Note that

$ \text{Ker}( \pi \circ \varphi ) = I $, as $ \text{Ker}( \pi ) = J $ and $ \varphi(I) = J $
$ \text{Im}( \pi \circ \varphi ) = R/J $, as $ \pi $ is surjective and $ \varphi \in \text ...

Let $X$ be a smooth projective surface over $ \mathbb{C} $ and let $H$ be an ample divisor on $X$. Let $L$ and $M$ be two $\mathbb{Q}$-divisors on $X$ which are not numerically trivial, and such that \[ L^2 = L \cdot M = M^2 = 0 \]Show that $L$ and $M$ are numerically proportional, i.e. \[ \exists ...