Questions about projective space in geometry, a space which can be seen as the set of lines through the origin in some vector space. As such it is a special case of a Grassmanian. See https://en.wikipedia.org/wiki/Projective_space

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On Kapranov's “On the derived categories of coherent sheaves on some homogeneous spaces”

As a graduate master student I am reading Kapranov's paper "On the derived categories of coherent sheaves on some homogeneous spaces" (1988). One problem is that the paper assume lot of notations and ...
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1answer
50 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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1answer
30 views

tangent space of a curve in projective space

Suppose we have the curve $Z\subset \mathbb{P}^2$ given by the equation $y^2z-x^3=0.$ I have to find a basis for the tangent space at $(0:0:1)$, but I find the definition hard: Let $X$ be a variety ...
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What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
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1answer
60 views

Hilbert function and homogenous polynomials.

Let $\{[1:0:0],[0:1:0],[0:0:1],[1:1:1] \} = \{p_1,p_2,p_3,p_4\}$ be four points in the projective space $\mathbb{P}^2$. For every $p_i$, show there is a homogenous polynomial $f_i$ such that ...
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Constructing Incidence variety without using equations

Let $k$ be a field. Let $X$ be the Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a specified Hilbert polynomial. Let $Y$ be another Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a ...
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1answer
32 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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28 views

Using Riemann-Roch

I have canonical divisor $K$ of curve $\mathbb{P}^1$, and I would like to find $l(2K)$, $l(3K)$ and $l(-K)$ using Riemann-Roch theorem. I know that $g=0$ in this case, so $deg(K)=2\cdot 0-2=-2$. ...
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23 views

Hyperplanes without Axiom of Choice

For any projective space that contains more than one point, is it possible to prove that it contains a hyperplane without using the Axiom of Choice? It's easy enough to prove that there exists a ...
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29 views

Understanding functions in $\mathbb{P}^1$

I just need solution check: I am given function $f\in k(\mathbb{P}^1)$ that sends $(x:y) $ in $\frac{x}{y}$. If I understood this right, this is just coordinate function $x$ since it send and point ...
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pencil of cubic curve passing six points

Let [1,0,0],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3] be a six points in general position. The question is how can determine the pencil of cubic curve passing through these points? Many thanks.
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projective nullstellensatz proof

I have to give a talk about projective varieties, including the projective nullstellensatz. As I'm not really into algebra or algebraic geometry, I've got some problems with the proof. Projectiv ...
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1answer
153 views

Group acting on a Projective Space

Let $G$ be an algebraic (zariski closed) subgroup of $SL(n,C)$ for some algebraically closed field $C$. Now $G$ acts on an $n$-dimensional vector space $V$ over $C$ where $V$ is a solution space of a ...
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19 views

Is the cone of a manifold a manifold of dimension one higher?

I think the cone of a manifold in complex projective space (the preimage of it by projection) would be a manifold of dimension one higher, but I don't know how to show this.
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16 views

Projection on the subspace perpendicular to a vector

I am running an algorithm and during one of the steps of the algorithm, I have to update a matrix $B$ but projecting every column of the matrix, $B_j$, j=1,...,k, on the subspace perpendicular to a ...
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1answer
71 views

Lines on a Quintic Threefold

We work over an algebraically closed field $k$. I've been given the exercise of showing (using only the technology introduced in the first chapter of Shafarevich's Basic Algebraic Geometry) that ...
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Rings of Regular functions, and regular maps between Quasi Affine to Quasi Proj. Varieties.

I have studied classical algebraic geometry a while ago. I want to sum up in short as possible everything regarding their rings of regular functions. If my understanding not correct, please correct ...
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29 views

describing proj. seurface.

I have the surface $W=Z(x_0x_1-x_2x_3)$, in $\mathbb{P^3}$ and I want to describe it as a union of an affine piece and some other piece laying in $\mathbb{P^2}$. My solution is to look at: ...
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Intersection of circle and line.

Now we can realize that intersection of two parallel lines at point of infinity in projective space. I can manage to visualize this and compute it. My issue is if we have the two varieties $ X_1= Z ...
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44 views

Projection Matrix related problem

Two orthogan projection operators $P_A$ and $P_B$ are said to be orthogonal if $P_AP_B=0$. This is denoted as $P_A \perp P_B$. Show that: a) $P_A$ and $P_B$ are orthogonal if and only if their ...
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49 views

Show that any two distinct lines in $\Bbb P^2$ intersect in one point.

Show that any two distinct lines in $\Bbb P^2$ intersect in one point. Proof(My attempt). Let $L_1, L_2$ be any two distinct lines in $P^2$. Write $L_i = V (a_iX + b_iY + c_iZ), i = 1,2$. It ...
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41 views

Localization of the ideal of some projective variety

Let $X \subset P^n$ be some projective variety, and $I$ be its ideal. We consider the 0th graded piece of the localization $(I \cdot k[Z_0,...,Z_n,Z_i^{-1}])_0$ in the coordinate ring ...
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1answer
33 views

All translations on $\mathbb{R}$ are induced by a $2\times2$-matrix

I want to prove that all translations on an affine straight line are induced by projective transformations of the projective extension. If we look at $\mathbb{R}$ as our affine straight line, ...
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Examples of homogeneous polynomials that define $\mathbb{P}^n$

The complex projective space $\mathbb{P}^n$ is also a projective variety. According to Hartshorne, a projective variety is defined a s the zero set of a subset of homogeneous polynomials defined on ...
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Uniqueness of affine cone

Let $Z$ be a projective variety embedded into $\mathbb P^n$. Then we can define an affine cone over $Z$ as the inverse image of $Z$ under canonical map $\mathbb A^{n+1}\setminus0 \to \mathbb P^n$. I ...
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1answer
57 views

Why does a parabola have a single point at infinity?

Consider the parabola $V(zy-x^2) \subset \mathbb P^2$. This parabola has only one point at infinity which is $[0:1:0]$. On the other hand we sketch the parabola, we see that there are two asympotes, ...
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1answer
28 views

Smoothness of projective hypersurface

I'm trying to understand the question and answer here, but I don't quite follow what they're doing, so here is my take on it. The problem is to show that in $\mathbb R P^2$, given a homogeneous ...
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1answer
49 views

Surface in complex projective space

Let $\mathbb CP^3$ be projective space. Consider polynomial $f$ of degree $k$ satisfying $f(a \mathbb v)=a^k f(\mathbb v)$ for complex number $a$ and vector $v \in \mathbb C^4$. For example ...
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1answer
135 views

Is there an easy way to find the cohomology ring of the complex projective plane?

I am trying to find the cohomology ring of $\mathbb{C}\mathbb{P}^2$. But I don't know how. We know from the CW structure of $\mathbb{C}\mathbb{P}^2$ that the cohomology groups must be $\mathbb{Z}$ in ...
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A CW complex for $\mathbb{CP}^n$ so that $\mathbb{RP}^n$ is a subcomplex

Both $\mathbb{RP}^n$ and $\mathbb{CP}^n$ arise as particularly simple CW complexes: $\mathbb{RP}^n$ has a single cell in every dimension between $0$ and $n$, and $\mathbb{CP}^n$ has a single cell in ...
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Can a 4-d quadric in $\mathbb{P}_5$ contain a $\mathbb{P}_3$?

In a 5-d projective space $\mathbb{P}_5$ the so-called Klein quadric is defined as the points that satisfy the equation $x_0x_3 + x_1x_4 + x_2x_5=0$ - where $(x_0 : x_1 : x_2 : x_3 : x_4 : x_5)$ are ...
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Calculating the stiefel whitney class of Tangent Bundle of projective space

I was reading the proof of the fact that whitney sum of the tangent bundle of $RP^n$ with the trivial bundle is isomorphic to whitney sum of $n+1$ tautological line bundles on $RP^n$ from Hatcher's ...
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Elliptic curve Schoof algorithm, projective polynomial point coordinates

I'm trying to understand Schoofs algorithm for determining $\#E(F_P)$ of an Elliptic curve $y^2 = x^3 + ax + b$ over $F_P$. For this I'm looking at the implementation of MIRACL: ...
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Resolution of the $E_8$ singularity with a weighted blowup

I am reading Miles Reid's notes on weighted projective spaces, and I'm a little confused about a particular paragraph (notes here, page 8): A famous case is the $E_8$ singularity $X: ...
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78 views

Translations on an affine straight line are projective transformations of the projective extension

How do I prove that translations on an affine straight line are induced by projective transformations of the projective extension? I know that a projective transformation is a projective map of a ...
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1answer
60 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
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205 views

Grouping Problem

Suppose there are 9 strangers. We will assign them into 3 groups and each group has exactly 3 people. For each grouping, the strangers who were assigned into the same group will get to know each other ...
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Why are projective coordinate rings not isomorphic when the corresponding projective varieties are?

I was trying to prove the following question from An Invitation to Algebraic Geometry by Karen Smith: Show that the homogeneous coordinate rings of projectively equivalent varieties are ...
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1answer
21 views

How to find the projection vector of an arbitary vector on a plane?

I have a 3D coordinate system defined by 3 perpendicular basis vectors (p), (q) and (r). On the other hand, I have an arbitary vector (d). I would like to find the vector (s), which is the projection ...
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1answer
18 views

Coordinate system in a Projective Space

How many coordinates does a point in $\mathbb{R}P^{n}$ have and which are the Homogenous Coordinates in $\mathbb{R}P^{n}$ ? To explain my question better. If we have a point p in the manifold ...
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1answer
52 views

How can I understand the isomorphism $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$ and compute the corresponding volumes?

I want to understand the above isomorphisms $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$. I seem to get some partial understanding but I miss a complete picture. For example I think that the last ...
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Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
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49 views

Self-intersection number in Projective Space

The question is based on the example given in Intersection Theory under the heading Self-intersection. The example is as follows: Consider a line $L$ in the projective plane $\mathbb{CP}^{2}$: it has ...
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1answer
33 views

Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
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40 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
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$U_{0}=\{[a^{0},a^{1},…,a^{n}]\in \mathbb{R}P^{n}\,:\,a^{0}\neq 0\}$. Is it open in $\mathbb{R}P^{n}$? [closed]

Is $U_{0}$ an open neighborhood of a $[a^{0},a^{1},…,a^{n}] \in \mathbb{R}P^{n}$ ? How can I prove that? It tried to see if $\pi^{-1}(U_{0})$ is open in $\mathbb{R}^{n+1}\smallsetminus\{0\}$. I see ...
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The degree of smooth projective curve included in $\mathbb P^n$ which is nondegenerate is more or equal the dimension of the projective space

I had a theorem during lecture, with proof which I don't understand. Theorem says: $ X \subset \mathbb{P}^n(\mathbb{C})$ smooth projective curve which is nondegenerate (not contained in a ...
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1answer
52 views

Number of points on a line in a finite projective plane

I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for ...
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Computing the inverse to a rational map

The setup: say I have some rational projective variety $X$ of dimension $n$ over $\mathbb{C}$ such that the map $$ X \dashrightarrow \mathbb{P}^n $$ is given by some linear series $\mathcal{L}$. My ...