3
votes
0answers
24 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
1
vote
0answers
26 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with Fubini-Study metric, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\overline w, \overline z] \qquad \eta ...
0
votes
0answers
28 views

A question about complex projective $n-$space

Let $\mathbb{P}^{n}(\mathbb{C})$ be the complex $n-$projective space and let $$ U_i=\{[x]=[x_0:\dots:x_n] \in \mathbb{P}^{n}(\mathbb{C}): x_i \ne 0\} $$ be a subset of $\mathbb{P}^{n}(\mathbb{C})$. I ...
4
votes
1answer
39 views

Closed subschemes of projective space

We work over the complex numbers. If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ ...
4
votes
1answer
94 views

some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
0
votes
0answers
15 views

About an automorphism of an algebraic curve.

Let $C \subset \mathbb{P}^2$ the Riemann surface given by the equation $X^6+Y^6+2Z^6=0$. Let $\phi:C \to C$ be the automorphism defined by $\phi([X:Y:Z])=[Y:X:Z]$ and consider $C/\tilde{}$, where ...
0
votes
1answer
71 views

Quotient of $\mathbb{CP}^1$ by antiholomorphic involution

On $\mathbb{CP}^1\ni(X_1:X_2)$ let $z=X_1/X_2$ be one of the two charts, and define an involution map $$I^-:z \mapsto -\frac{1}{\overline{z}}.$$ Question: how to prove that the quotient ...
4
votes
0answers
43 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
1
vote
1answer
74 views

(Complex) Projective Space

I followed a course in projective geometry and I'm not sure about 2 things: If I have 6 lines in projective space (IPĀ³) with commun secant, why are the 6 corresponding tensors linearly dependent? ...
4
votes
1answer
290 views

Pullback of very ample sheaf again very ample? And other questions.

Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
2
votes
2answers
105 views

Blow-up of $\mathbb{C}^2$ at $(0,0)$

I am reading a text in which, at some point, the author define the blow up of $\mathbb{C}^2$ at $(0,0)$ as "a complex manifold $\hat{\mathbb{C}}^2$ obtained by identifying two copies of $\mathbb{C}^2$ ...
3
votes
2answers
171 views

Lines on algebraic surfaces and duality

In the following, all varieties will be algebraic over $\mathbb{C}$. I have some general problems with concepts like the "space of lines in $\mathbb{P}^5$", "space of lines on a surface in some ...
4
votes
1answer
180 views

Proof of Moisezon Theorem

We call a compact complex manifold Moisezon manifold, if its dimension coincides with the algebraic dimension, i.e. it has as many algebraically independent meromorphical functions as its complex ...
4
votes
0answers
202 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?