# 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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### 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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### 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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### 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 ...
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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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### 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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