# Questions tagged [projective-varieties]

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32 questions
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### Projecting a projective variety away from a linear subspace

I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument. Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ ...
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111 views

### Automorphism Group of a Variety acts on Local Sections

The motivation/background of my question arises from following thread: Galois morphism - group acting on the variety The original setting is that we have a finite Galois morphism $f: X \to S$, where ...
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### Group Acting on Variety

My question refers to some comments occured in following thread: Galois morphism - group acting on the variety The setting is that we have a finite Galois morphism $f: X \to S$, where $X$ and $S$ ...
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### Galois Action on Coherent Sheaves Exact Functor

Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined ...
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### Degree and codimension of nondegenerate varieties

A projective variety $X \subset \mathbb{P}^r$ (i.e. reduced, irreducible closed subscheme) is called nonegenerate, if it is not contained in any hypersurface $H \subset \mathbb{P}^r$. Equivalently, ...
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### Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
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### Differential 1-forms on an irreducible projective variety

Let $k$ be an algebraically closed field and let $X$ be an irreducible projective variety over $k$. I am wondering what the module of differential 1-forms on $X$ is. Since $X$ is a projective variety,...
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### Shafarevich, locally regular $\Rightarrow$ globally regular

So I am confused with this argument in 3rd Ed, pg 47, Basic Alg. Geo. 1. Definition 1: if $X \subseteq \Bbb P^n$ is a quasiprojective variety, $x \in X$, and $f=P/Q$ is a homogenous function of ...
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### Is the Zariski topology on a variety $V$ a maximal Noetherian topology?

Let $K$ be an algebraically closed field. By a variety $V$ definable over $K$, I mean a quasi-projective or an algebraic variety in sense of Weil. It is the set of points in an affine or a projective ...
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