# Questions tagged [projective-varieties]

228 questions

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### Algebraicity of compact Riemann surfaces

I am taking a course in Riemann surfaces, in which the classical result about algebraicity of compact riemann Surfaces has been proven. However, I think there are some dubious points in the proof.
Here is the outline: let X be our compact connected…

Colard

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### Isomorphism between projective varieties $\mathbf{P}^{1}$ and a conic in $\mathbf{P}^{2}$

I'm trying to establish an isomorphism between the projective line $\mathbf{P}^{1}$ and the conic in $\mathbf{P}^{2}$ defined by $Y=Z(g)$, where $g=x^2+y^2-z^2$. This is part of exercise I.3.1 in Hartshorne's Algebraic Geometry.
I defined $\varphi…

JDZ

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### Irreducible components of exceptional locus.

Let $\phi:X\rightarrow Y$ be a birational regular map between projective varieties where $Y$ is non-singular. Define $C=\{q\in Y:\dim(\phi^{-1}(q))>0)\}$. Let $G=\phi^{-1}(C)$. I saw the following statement:
"Irreducible components of $G$ are…

Cusp

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### Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

Let $k$ be an algebraically closed field. All spaces are equipped with the usual Zariski topologies.
All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ intersect but this doesn't necessarily hold in…

Luigi M

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### Analytification of a smooth projective variety is a compact Kähler manifold.

I am reading “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrecht. On page 130 it is written that by Hodge theory there is a natural direct sum decomposition
$$H^n(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}$$ with…

Pouya Layeghi

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### Hartshorne Exercise I.7.7

I'm trying to solve the following exercise from Hartshorne's Algebraic Geometry, namely Exercise I.7.7
Exercise I.7.7: Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^{n}$. Let $P \in Y$ be a nonsingular point. Define $X$ to…

user940160

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### Branched cover in algebraic geometry

I've watched a lecture on K3 surfaces (although K3 surfaces are not the point of this question) where the following example is given:
Let $\pi:S\stackrel{2:1}{\to}\Bbb{P}^2$ be the branched double cover ramified over a smooth sextic curve…

rmdmc89

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### Double lines in $X$, implies $X \cong \mathbb{P}^{n}$

Let $X$ be a smooth, complex, projective variety. How to prove that if through two general points of $X$ there exists a double line, then $X \cong \mathbb{P}_{\mathbb{C}}^{n}$?

rla

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### Why is an elliptic curve with $j$-invariant in $K$ defined over $K$?

I often see in the literature some arguments like this:
"to show that an elliptic curve $E$ is defined over $\mathbb{F}_p$,
we show that the $j$-invariant of the curve is in $\mathbb{F}_p$."
But I thought in general there are elliptic curves not…

Andy

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### What is the meaning of the residue field of a point in scheme?

If I consider the analogy of local ring at a point to the space of function germs at the point, then the residue field can be seen as the values that functions can take at the point.
But when I consider the residue field of generic point or the…

Sayako Hoshimiya

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### How does gluing of affine patches of toric varieties at the examples $\mathbb{P}^2$ and $\mathcal{H}_r$ work?

I don't fully understand how the gluing of the affine parts of a toric variety exactly works. I have a hard time developing a common sense or any intuition how to tell the result of a gluing morphism immediately and I would appreciate any help in…

LegNaiB

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### Integral cohomology. follows from GAGA?

Let $X$ be a projective variety over $\Bbb C$. One can compute the integral cohomology groups $H^p(X,\Bbb Z)$ by looking at the constant sheaf $\Bbb Z$ in the Zariski topology on $X$, but one can do the same with respect the Euclidean topology on…

Hammerhead

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### The projective closure of the twisted cubic curve

I'm now reading Hartshorne's Algebraic Geometry and trying to solve Exercise 2.9(b).
Let $Y$ be an affine variety in $\mathbb{A}^n$. Identifying $\mathbb{A}^{n}$ with the open subset $U_0$ of $\mathbb{P}^n$ by the homeomorphism $\varphi_{0}:…

Hetong Xu

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### How to understand group action especially Galois action on a scheme?

I read a lot of books and find none of them give explicit descriptions of group action on schemes. I am very confused now and have lots of questions. So I think these questions will be relatively long and hope you could read them.
First, in many…

Mike

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### Identifying $\mathbb P^L$ with the space of $N+1$- tuples of homogeneous polynomials of degree $d$ in $N+1$ variables

Let $k$ be a field. $N,d$ be positive integers and define $L= {N+d \choose d } (N+1)-1 $ .
Then can we identify $\mathbb P^L$ with the space of $N+1$- tuples of homogeneous polynomials of degree $d$
in $N+1$ variables , with at least one…

uno

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