Questions tagged [projective-varieties]
346 questions
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What is Chow's lemma really about?
By Chow's lemma, I mean any variant of the following basic result in algebraic geometry relating complete varieties to projective varieties:
Lemma.
For any complete variety $X$, there exist a projective variety $\tilde{X}$ and a surjective…
Zhen Lin
- 97,105
9
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0 answers
Explicit example of a quintic threefold with 2875 distinct lines
A generic quintic threefold has 2875 lines. Is there an example of a quintic threefold explicitly defined by a polynomial that can be proven to have exactly 2875 distinct lines?
I am particularly interested in an answer in the following form.…
Weiyan Chen
- 111
9
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2 answers
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
- 2,265
9
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4 answers
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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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…
user880214
6
votes
1 answer
Why are meromorphic functions on a smooth projective curve rational?
Let $C \subset \mathbb P^n$ be a smooth connected projective curve over $\mathbb C$. Then the function field $k(C)$ consists of all functions $f$ which can locally (in the Zariski topology) be written as quotients of polynomial functions on $C$.
On…
red_trumpet
- 11,169
6
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2 answers
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…
6
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0 answers
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
- 218
6
votes
3 answers
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
- 4,053
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Exact sequence obtained from a generic pencil on a projective surface
Let $X$ be a normal projective surface with only quotient singularities. I have a question about the proof of Theorem 10.8 (p.119) in Megyesi's Chern Classes of $\mathbb{Q}$-sheaves. It goes as follows:
Fix a projective embedding of $X$. A generic…
blancket
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5
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Is Every Closed Algebraic Set of Dimension $n$ Contained in a Closed Variety of Dimension $n+1$
Let $V$ be an algebraic variety of dimension $m$ over an algebraically-closed field of characteristic $0$, and let $n
Thomas Anton
- 2,446
5
votes
1 answer
Hartshorne Chapter 1 Exercise 7.7 (a)
I am trying to solve part (a) of the following exercise of Hartshorne:
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 be the closure of the union of all lines $P Q$,…
Eric
- 531
5
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0 answers
Space of simple tensors
A simple tensor in $V^{\otimes n}$ is one that can be written as $v_1 \otimes \cdots \otimes v_n$ for some choice of $v_i \in V$, these are also called rank 1 tensors. The space of these simple tensors $V^{\otimes n}_{\text{rank} = 1}$ is not closed…
QCD_IS_GOOD
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5
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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
- 1,500
5
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0 answers
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
- 861