Questions tagged [projective-varieties]

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Normal bundle of fibers

Everything is defined over the field of complex numbers. Let $Y \subset \mathbb P^{2n+1}$ be a smooth projective variety of dimension $n$. Denote by $E$ the (naive) projectivization of the normal ...
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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 ...
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Find the equations of a curve under a morphism

We work over the field of complex numbers. Consider the curve of degree $(1,5)$ inside $\mathbb{P}^1\times \mathbb{P}^1$ defined as $$C: ux^5+vy^5=0,$$ where $((u,v),(x,y))$ are the homogeneous ...
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How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$

I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation: \begin{equation} z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3 \end{equation} Let $x_i, i=0,1,2$ denote the ...
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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 ...
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multiplicity of the intersections between $p=x_0x_1^2+x_1x_2^2+x_2x_0^2\,\,$ and $\,\,q=-8(x_0^3+x_1^3+x_2^3)+24x_0x_1x_2$ in $\mathbb{P^2(K)}$

in $\mathbb{P^2(K)}$ where $\mathbb{K}$ is an algebraically closed field and $[x_0,x_1,x_2]$ the homogeneous coordinates, consider the following (homogeneous) polynomials: $p=x_0x_1^2+x_1x_2^2+x_2x_0^...
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Real Projective plane $\mathbb{RP}^2$ [closed]

I visited this site about Real Projective plane $\mathbb{RP}^2$, or $\mathbb{P}^2\bigl(\mathbb{R}\bigr)$ if you prefer. My problem is this: when I implement the first equation of the Cross-capped disk ...
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What is a generically reduced scheme?

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30) Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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Describing all endomorphisms of elliptic curves

Let $k$ be an algebraically closed field. A curve is a separated integral $k$-scheme of finite type over $k$ and of dimension one. An elliptic curve $E$ is a smooth projective curve of genus one (a $k$...
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1 answer
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Degenerate plane conics form closed subset in $\mathbb{P}^5$ isomorphic to projection of $\mathbb{P}^2\times\mathbb{P}^2$

I am doing the following problem (Exercise 5.3.5) from "An Invitation to Algebraic Geometry" by Karen E. Smith et al.: Show that the subset of all degenerate plane conics naturally form a ...
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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. ...
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1 answer
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Isomorphism between $Z(x_1^2-x_0x_2) \subseteq \mathbb{P}^2$ and $\mathbb{P}^1$

I want to show that if I have morphism $\varphi_0 : C_0 \rightarrow \mathbb{P}^1, \varphi_0([a_0:a_1:a_2])=[a_0:a_1]$ then I can extend it to be an isomorphism $\varphi : C:=Z(x_1^2-x_0x_2) \...
1 vote
1 answer
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Projection of rational normal curve is still a rational normal curve (of smaller degree)?

We work over $\mathbb{C}$. Let us define the rational normal curve of degree $d$ as the image of the morphism $$\nu_d:\mathbb{P}^1\to \mathbb{P}^d,\quad [x:y]\mapsto [x^d:x^{d-1}y:\ldots:xy^{d-1}:y^d]....
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Calculate the Vanishing Ideal of the Image of a Variety under Projection Map

Let $X=V(I) \subset \mathbb{P}^n$ be a (complex) projective variety described as vanishing set of an ideal $I= \langle f_1,..., f_n \rangle$ (the $ f_1,..., f_n$ are appropriate homogeneous ...
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Fano Variety of a Hyperplane (Harris Algebraic Geometry A First Course)

I have a question about Example 6.19 (p. 70) in Harris' Algebraic Geometry: A First Course. There is stated that the Fano variety $$F_k(X) := \{\Lambda \in \mathbb{G}(k,n) \ \vert \ \Lambda \subset X \...
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Space of sections of adjoint varieties

We will work over the field of complex numbers and we will be consistent with the Grothendieck porjectivization, that is $\mathbb P(\cdot)=\operatorname{Proj}(\operatorname{Sym}(\cdot))$. Let $X$ be ...
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Proof of Noether-Castelnuovo's Theorem using elementary links

Let $\chi:\mathbb P^2\dashrightarrow\mathbb P^2$ be a birational map of the plane. Then $\chi$ is the composition of linear automorphisms and of the standard Cremona map : $(x:y:z)\mapsto(\frac{1}{x}:\...
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What is a k-tangent variety?

Introduction I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my ...
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Kempf's Proof that Projective Varieties are Complete

The following proof is taken from "Algebraic Varieties" by Kempf. There seems to be way too many $n$'s in the proof. The projective variety is $\mathbb{P}^n$, the number of polynomials is $...
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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 ...
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How to prove this is an isomorphism of varieties

So I’m trying to prove all conics (i.e. zero sets of irreducible homogeneous polynomials of degree $2$) in $\mathbb{P}^2$ are isomorphic to $\mathbb{P}^1$ (here I work with classical algebraic ...
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Explicit computation for an elliptic curve.

Let $C\subset \mathbb P^2$ be the smooth curve $Y^2Z=X^3+Z^3$ and let $p:C_0\to \mathbb A^1$ be the projection $(x,y)\mapsto x$ from the affine part $C_0$ of $C$ (described by $y^2=x^3+1$) onto the $x$...
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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 $\mathbb{F}_p$." ...
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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 ...
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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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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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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 ...
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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 ...
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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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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. ...
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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 $\pi$ ...
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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 ...
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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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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 ...
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3 votes
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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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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 ...
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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 ...
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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 $X$ ...
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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 vote
1 answer
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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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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 \...
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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. ...
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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)\...
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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 ...
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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 \}:= \...
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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 ...
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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 $k$. ...
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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 ...
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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 ...
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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})=\...

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