# Tag Info

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### Branched cover in algebraic geometry

Let's tackle ramified/unramified first, since that's something that's pretty uniform across the literature. Definition (ref): A morphism of schemes $f:X\to S$ is unramified at $x\in X$ if there ...
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### Analytification of a smooth projective variety is a compact Kähler manifold.

Yes, this boils down to two facts, which you should be able to find in e.g. Huybrecht's Complex Geometry or Voisin's Hodge Theory and Complex Algebraic Geometry: I. $\mathbb{P}^n(\mathbb{C})$ is a ...
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### Why are meromorphic functions on a smooth projective curve rational?

That’s a good question! Note that it is a (very) special case of the GAGA theorems/philosophy – in ”complete enough” situations (usually properness), analytic objects and morphisms should come from ...
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### Clarification about Ideal and zero sets of empty set in Varieties

There are many reasons, but I guess what I would think as the most important one is that it is the only definition for which $S\subseteq T \implies Z(T)\subseteq Z(S)$ (and the obvious similar ...
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### The projective closure of the twisted cubic curve

First we can verify that the image of $\overline{v}$ is closed: it's the vanishing locus of $xw=yz$, $xz=y^2$, and $z^2=xw$. Thus the image of $\overline{v}$ is a closed set containing $Y$, and it's ...
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### Relation between tautological line bundle and blow up at the origin

The way these definitions "talk to each other" has to do with the two projections onto the factors. If you map to $\mathbb C^{n+1}$, you get the blowup. If you map to $\mathbb P^n$, the same ...
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### universal property of Albanese variety

This is the definition of the Albanese variety. Presumably you mean something like why the dual of $\mathrm{Pic}^0$ is the Albanese variety in good situations? I've always liked the appendix to this ...
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### All varieties are quasi-projective?

That assertion is not a theorem, it is just informing you that those are the only varieties consideredin Chapter I and that's what you should understand when you read the word variety within that ...
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### Algebraic, Projective, and Riemannian Geometry: How do they interact?

Let me try to answer your question to some extent; given the vague nature of the question, there will be no canonical answer. You should think in terms of three different areas of mathematics: ...
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### Rational points on projective varieties: what is density?

A scheme $X$ has an underlying topological space $|X|$. When one asks whether the $k$-points are dense, one usually refers to density as a subset of $|X|$.
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### Intersection between projective curves

Bezout's Theorem depends on the field you are working over. Roughly, the Theorem states that if you have curves of degree $d$ and $e$, then the curves will intersect in exactly $de$ points in the ...
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### Isomorphism between projective varieties $\mathbf{P}^{1}$ and a conic in $\mathbf{P}^{2}$

1) The isomorphism $\phi$ is between $\mathbb P^1$ and a conic $Y\subset\mathbb P^2$, not a cone. (This conic is related to the cone in $\mathbb A^3$ with the same equation, but you should not ...
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### Existence of a hyper surface containing the singular locus of projective variety.

The following more general fact is true. Proposition: For any closed subset $S \subset \mathbf P^n$ and any point $p \notin S$ there is a hypersurface $W$ which contains $S$ but not $p$. (Your ...
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### Showing the join of two disjoint projective varieties is a projective variety.

From my reading, it appears that the assumption of disjointness is used in defining the "join" construction, not the proof of showing that it is projective. Indeed, if $X\cap Y$ is nonempty, then the ...
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### There is a line in $\mathbb{P^2}$ not passing through any of a finite collection of points
I’m assuming the base field is infinite, otherwise it can be a little trickier to state. This is equivalent to the statement: for any $p_1,\ldots,p_s \in \mathbb{A}^3 \backslash \{0\}$, there are \$a,b,...