# Tagged Questions

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### 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 ...
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### 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 ...
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### (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? ...
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### 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 ...
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### 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 $\mathbb{C}^2$ ...
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### 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 ...
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 $\mathbb{R}_{+}$?