# Questions tagged [projective-varieties]

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• 23
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### Is there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$

I'd like to ask that if there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$. I cannot think of an example.
• 277
1 vote
54 views

### How do I compute the action of an automorphism on the Néron-Severi group of a projective variety?

I am trying to read Dynamics of Automorphisms of Compact Complex Surfaces by Serge Cantat, and I am confused by his example of surfaces of degree (2, 2, 2) in section 2.4.6. The setup is the following:...
• 113
67 views

### classification of morphism between varieties

Let $K$ be an algebraically closed field. We have different important theorems to study morphism of variety, in particular there is a clear and constructive description of the morphism between affine ...
• 717
1 vote
119 views

### If $f:X\rightarrow Y$ is a $k-$morphism of projective varieties and $X(k)\neq\varnothing$, then $Y(k)\neq\varnothing$

Suppose $X\subset \mathbb{P}^m,Y\subset\mathbb{P}^n$ are projective varieties defined over a field $k$, and that $f:X\rightarrow Y$ is a $k-$morphism, i.e. $f$ is a morphism which induces a $k-$...
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### Finite morphism vs morphism of finite degree

Let $f:X \to Y$ be a morphism of projective varieties (integral schemes of finite type) over a field $k$. If $f$ is dominant, then $f$ induces a field extension $K(Y)\to K(X)$ of function fields. We ...
• 560
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### The canonical sheaf of hypersurface on $\mathbb{P}^n$

I want to prove the following result Let $X\subset\mathbb{P}^n$ be a smooth projective variety defined over $k.$ Let $Y= V(f)$ be a smooth subvariety of $X$ defined by a homogeneous polynomial $f$ of ...
• 41
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### Defining algebraic varieties in general

I have encountered two general notions of algebraic variety when reading different texts in algebraic geometry, and wanted to ask whether they were equivalent or whether one is stronger than another. ...
• 404
1 vote
113 views

• 301
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### Composition of Gysin and restriction maps on $\ell$-adic cohomology

I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145. Let $k$ be an algebraically closed field, ...
• 5,494
121 views

### Does the completions $\hat{\mathscr{O}}_P(X)\simeq \hat{\mathscr{O}}_Q(Y)$ could deduce local rings $\mathscr{O}_P(X)\simeq \mathscr{O}_Q(Y)$?

In Hartshorne's algebraic geometry, he said the completion of local ring $\hat{\mathscr{O}}_P(X)$ takes much more 'local properties' than the local ring $\mathscr{O}_P(X)$. There are two natural ...
• 141
114 views

### Compute the Zariski closure of a set

I have to compute the Zariski closure of the image of the following rational map: $f:{P}^2 \rightarrow \mathbb{P}^4$ $[x_0:x_1:x_2]\rightarrow [x_0x_1:x_0x_2:x_1^2:x_1x_2:x_2^2]$ I have already proved ...
• 117
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• 1,115
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### Generator of the maximal ideal of $\mathcal{O}_p$

I know that $x^3-xz^2-y^2z=0$ is a nonsingular curve in $\mathbb{P}^2$(char $k\neq 2)$. By definition, each local ring is a regular local ring. Consider the ring of degree zero elements in a ...
• 453
68 views

### Is a nonconstant morphism of projective varieties necessarily finite?

Let $k$ be a field, and let $X$ and $Y$ be projective varieties over $k$. Do there exist morphisms $X \to Y$ that are neither finite nor constant? I know this cannot happen for $X$ and $Y$ curves (as ...
• 560
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• 874
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### How to prove the image of the morphism $\varphi:C\to \mathbb{P}^1$ is dense in $\mathbb{P}^1$?

Consider an affine variety $C$ in $\mathbb{P}^2$ determined by $x^2+y^2-z^2=0$, then how to prove the image of the following morphism is dense in $\mathbb{P}^1$? \begin{align} \varphi:C&\to \...
• 141
1 vote
156 views

### Applications of the Lefschetz Hyperplane Theoren

There are a couple of applications of the Lefschetz Hyperplane Theorem I am struggling to wrap my head around. Hopefully someone knows how these facts are deduced directly from the theorem. Suppose $X$...
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