Questions tagged [projective-varieties]

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4
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1answer
83 views

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 $\mathbb{F}_p$." ...
3
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0answers
40 views

Support of homology in quasi-projective varieties.

Given a quasi-projective complex variety $X$ and a positive integer $i<\text{dim}(X)-1$. Consider the homology group $H_i(X(\mathbb{C}))$. Is it possible to find a subvariety of codimension at ...
1
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1answer
49 views

Two affine varieties which are isomorphic, but their projectivisations are not

I am learning Algebraic Geometry and came across the following question: Show $V(y-x^3) \cong V(y-x^2)$ as affine varieties in $\mathbb{A}^2$. Prove that their projectivisations in $\mathbb{P}^2$ are ...
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1answer
47 views

Proof of (slight generalization of) Chow's lemma for varieties

In Algebraic Varieties, Kempf proves a slight generalization of Chow's lemma (in the case of varieties): letting $Y$ be an irreducible separated variety, there exists an irreducible projective variety ...
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0answers
48 views

What is an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$?

It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$. I want to understand this with a lot of examples. For example, let $a,b\in K^\times$ and ...
2
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0answers
29 views

Differential of the projectivization of V following Shafarevich

In II.1.3 of Shafarevich's book Basic Algebraic Geometry 1, in page 90, it is written the following: However, I understand that the differential is given by a precomposition, so I think that it ...
0
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0answers
18 views

Reason for a surface to be minimal

Let $B$ and $S$ be smooth irrational curves, and $G$ a group acting faithfully on $B$ and $S$, such that $B/G$ is elliptic and $F/G$ is rational. Why is that true that $S=(B\times F)/G$ is minimal? ...
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78 views

Exercise 8.2 Harris Algebraic Geometry

I am unfortunately unable to solve the following exercise: Let $\Lambda_1,\ \Lambda_2$ be two-planes in $\mathbb{P}^4$ meeting in one point $p$, and let $C_i\subset \Lambda_i$ be conic curves. ...
0
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0answers
29 views

Pullback of a rational curve

Let $F$ and $H$ be curves, and $G$ be a finite group acting faithfully on $F$ and $H$. Denote $\pi$ the canonical projection from $F\times H$ on $(F\times H)/G$, and suppose that this morphism $\pi$ ...
1
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1answer
24 views

Embeddings of a surface that are given by linear systems of divisors

I am reading the proof of Castelnuovo's contractibility criterion in Beauville's Complex Algebraic Surfaces. I would like to clarify a paragraph. We have a hypeplane section $H$ of a surface $S$, a ...
0
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1answer
33 views

Embedding of a variety

Let $D$ be a divisor on a variety $X$, and assume that $h^0(X,\mathcal O_X(D))=n+1$. So let $s_0,\dots,s_n$ be a basis of $H^0(X,\mathcal O_X(D))$. I am trying to understand the following statement : &...
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0answers
51 views

Are complete varieties in the sense of Weil the same as proper varieties?

This is a continuation of a previous question; I will use the same notation and terminology. An abstract variety (or Variety, with a big V) in the sense of Weil consists of the following data: A ...
3
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1answer
40 views

The tensor product of the canonical line bundle and k(x) for a closed point x

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. At the beginning of the page 91, it is written that if $X$ is a smooth projective variety with a canonical ...
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0answers
82 views

Existence of a certain polynomial on $\mathbb P_k^n$

Let $k$ be an infinite field. Let $X\subseteq\mathbb P_k^n$ be a projective subvariety. Let $D$ be a Cartier divisor on $X$. Mumford and Oda’s Algebraic Geometry II says (in the paragraph following ...
3
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1answer
193 views

Algebraic, Projective, and Riemannian Geometry: How do they interact?

The aim of this question is to understand the interaction between projective algebraic varieties (over the complex or real numbers), Riemannian manifolds, and projective space, through shared concepts ...
0
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1answer
48 views

Morphism between a projective and an affine varieties has finitely many points in its image

I was asked to show that if $X$ is a projective variety, $Y$ an affine variety and $\varphi:X\rightarrow Y$ is a morphism, then $\varphi(X)$ is finite. I think that I have to use this fact: Let $X$ ...
1
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1answer
50 views

Map from a curve to the projective plane

I am working on a exercise and I would like to ask you for your help because I am terrible lost. Let us suppose that $C\subset \mathbb{P}^2_k:=\text{Proj}(k[x_0,x_1,x_2]$ is an algebraic curve given ...
1
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1answer
62 views

What is the moduli space $M _{0,5}$?

Let $M_{0,n}$ denote the moduli space which consists of genus 0 non singular projective curves with n distinct marked points upto marked point isomorphism. So $M_{0,3}$ is a singleton set as any 3 ...
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33 views

What is the Zariski Closure of $C=\{[x:y:z] \in \Bbb P^2 \mid x=1,y=0\}$ in $\Bbb P^2$?

I want to determine the Zariski Closure of $C=\{[x:y:z] \in \Bbb P^2 \mid x=1,y=0\}$ in $\Bbb P^2$. My guess is that $\bar C=\{[x:y:z] \in \Bbb P^2 \mid y=0\}$, for this assume that $E=\{[x:y:z] \in \...
0
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1answer
58 views

Divisor class group of $X$ and $X - p$

Let $V$ be a projective variety in $\mathbb P^n$ and $X=C(V)\subset \mathbb A^{n+1}$ its affine cone. Let $\bar X$ be the projective closure of $X=C(V)$ and let $P$ be the vertex of the cone. ...
0
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1answer
31 views

Quartic in the projective space and exact sequence of sheaves

I am trying to understand a few basics about the twisting sheaves. I read that, given a smooth quartic $S$ in $\mathbb P^3$, we have an exact sequence $0\rightarrow\mathcal O_{\mathbb P^3}(-4)\...
1
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1answer
88 views

Very Ample sheaves

Let $X$ be the non singular cubic curve $y^2z=x^3-xz^2$ in projective space of dimension 2. Let $L$ be the invertible sheaf $L(P_0)$. How does $L(P_0)$ not being generated by global sections imply ...
0
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1answer
86 views

How do we construct the projective line?

According to Example $5.5)$ on the following PDF, projective line $\mathbb{P}^{1}$ is obtained from the following process. And the author defines $\mathbb{A}^1 \cup \left \{ \infty \right \}:= \...
4
votes
2answers
168 views

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 ...
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0answers
61 views

If $X$ is a smooth projective curve and $g(X)=0$, then $X$ is a plane conic.

I've read the following statement in Manin-Tsfasman's survey, Proposition 1.1.1 (I'm sorry I could only find the link in Russian): Let $X$ be a smooth, projective rational curve over a field $k$. ...
1
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1answer
149 views

Smoothness implies a condition on the Jacobian in every affine open

In chapter 12 of FOAG, Ravi Vakil defines smoothness in the following way A k-scheme is k-smooth of dimension d, or smooth of dimension d over k, if it is of pure dimension d, and there exists a ...
0
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1answer
67 views

Transitivity of being parabolic subgroup

Let $G \subseteq GL_k(V)$ be an affine algebraic group. Hence $k$ is a field, $dim_k(V)=n$ and $G \subseteq GL_k(V)$ is a closed subgroup. A closed subgroup P is called parabolic if $G/P$ is a ...
4
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1answer
68 views

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})=\...
0
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1answer
39 views

Very ample line bundle and the induced embedding

Over $\mathbb C$, let $X$ be a projective variety and let $\mathcal L$ be a very ample line bundle. Then there is an induced embedding $X\to \mathbb P(V)$ for $V=H^0(X;\mathcal L)^*$. It is easy to ...
0
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1answer
101 views

Hartshorne Chapter 1 exercise 6.4: Maps of curves and function fields.

I'm solving problems in Hartshorne. I don't know how to solve the following exercise(6.4 of Chapter 1): Let $Y$ be a nonsingular projective curve. Show that every nonconstant rational function $f$ on $...
1
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0answers
111 views

Generators of the ideal of the projective closure

Let $X=\{(t,t^2,t^5): t\in k\}\subset \mathbb{A}^3$. Show that the projective closure $\bar{X}$ is a projective variety of dimension 1, and say if it is isomorphic to $\mathbb{P}^1$. Compute $\mathbb{...
5
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0answers
121 views

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 $\mathbb{P}^{n}$. ...
1
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1answer
81 views

How many blowups do we need to make a pencil base point free?

Let $\Bbb{P}^2$ be the projective plane over $\Bbb{C}$. Take a pencil of curves of degree $d$ on $\Bbb{P}^2$ given by a dominant rational map $\phi:\Bbb{P}^2\dashrightarrow \Bbb{P}^1$. The pencil has $...
2
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1answer
65 views

Arrow-reversing Proj and blow-up

Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ ...
0
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1answer
25 views

Let $V,W$ be quasi-projective varieties. Then $f:V\rightarrow W$ is regular iff $f_i:f^{-1}(W_i)\rightarrow W_i$ are regular for all $i.$

I want to prove the following property: Let $V,W$ be quasi-projective varieties, let $W=\bigcup\limits_{i\in I}W_i$ be an open covering and let $f_i:f^{-1}(W_i)\rightarrow W_i$ be induced by $f:V\...
0
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1answer
39 views

Direct image of a vanishing set $V_+(H(\overline{X}))$ expressed as vanishing set of polynomials

Let $r: \mathbb{P}^n_{\mathbb{C}} \dashrightarrow \mathbb{P}^n_{\mathbb{C}} $ be a birational map $$ [X_0: X_1: \cdots : X_n] \mapsto [F_0(\overline{X}): \cdots F_n(\overline{X})]$$ where $F_i(\...
3
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0answers
82 views

Why the affine homography matrix can be recovered from the combination of elements of largest eigenvalues?

In the book "Multiple View Geometry in Computer Vision" of R. Hartley, A. Zisserman there is an Algorithm 4.7, describing how to find affine homography. I give link the to the book later, ...
0
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2answers
67 views

Example of morphism of irreducible varieties with reducible fibre

Suppose $f: X \to Y$ is a morphism of irreducible affine or projective varieties. Does it then always follow that $f^{-1}(p)$ is irreducible for any point $p \in f(X)$? I think it's not true, and I'm ...
-1
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1answer
35 views

Support of principal divisor and exceptional locus

Assume we have a birational morphism between smooth varieties $\phi : X \longrightarrow Y$. Let $f$ be a non-zero rational function of $X$, and $E$ be the exceptional locus of $\phi$. I think we ...
1
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2answers
64 views

There is a line in $\mathbb{P^2}$ not passing through any of a finite collection of points

I'm working on Fulton's Algebraic Curves and having a little bit of trouble with an exercise from Chapter 4 on projective varieties (4.25). The question is as follows: Let $P = [x:y:z] \in \mathbb{P}^...
0
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1answer
53 views

All varieties are quasi-projective?

In Hartshorne's book "Algebraic geometry", page XV ("Terminology") says: "all varieties in Chapter 1 are quasi-projective". Could anyone give me a detailed explanation ...
2
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1answer
55 views

A nonsingular plane curve of degree 5 has no linear system of dimension 1 and degree 3

Let $C$ be a nonsingular plane curve of degree 5 over $\mathbb{C}$, as in the question, I want to show that $C$ has no linear system of dimension 1 and degree 3, that is has no $g^{1}_{3}$. First, ...
1
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1answer
55 views

How to show the kernel of the map is following?

Let $X$ be a smooth projective variety over $\mathbb{C}$, $\mathcal{L}$ be a line bundle on it. Let $s,t$ be two elements of $H^{0}(X,\mathcal{L})$, that is $s,t$ are global sections of $\mathcal{L}$. ...
1
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0answers
74 views

Proposition 18.10 from Joe Harris' Algebraic Geometry

I'm trying to understand a step in Harris book in proof of a proposition (Algebraic Geometry by Joe Harris, p. 230) which looks quite confusing: Proposition 18.10. Let $X \subset \mathbb{P}^n$ be an ...
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0answers
100 views

Non-Degeneracy of Ration Normal Curve

We consider the so called 'Steiner construction', which can be looked up e.g. in this script e.g. which gives us an irreducible curve. The construction works as follows: Let say we have $\Lambda_1, ......
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0answers
135 views

Steiner Construction of Rational Normal Curve

I learned in this script on Geometry of Algebraic Curves by Joe Harris that a rational normal curve can be constructed by a nice, old-fashioned construction, called the Steiner construction (page 79). ...
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0answers
41 views

Minimal embedding for blowing ups

Let us consider the following specific problem for blowing-ups. Let $n$ be a large positive integer. Let $X\subset \mathbb P^n$ be a smooth sub variety of codimension $>1$. Denote $Y$ the blowing-...
3
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1answer
48 views

Complete intersection curve is a Riemann surface - Miranda Prop. 3.9

I am trying to to prove Proposition 3.9 from Miranda's book "Algebraic Curves and Riemann Surfaces" which he gives without proof. Overall the proof mainly involves a use of the implicit ...
0
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1answer
89 views

Computing the image of a projective variety

Let $f$ be an endomorphism of projective space $\mathbb{P}^n$. From this answer I know that $f$ is proper. But how does one actually determine the image of a given arbitrary Zariski closed subset? For ...
0
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1answer
48 views

Extrinsic vs intrinsic definitions of classical varieties

In differential geometry, it seems that a distinction is usually made between extrinsic and intrinsic definitions of real manifolds. The former is as a space defined by equations of a "nice ...

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