for questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

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10
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3answers
1k views

$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
10
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2answers
478 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
10
votes
3answers
264 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
8
votes
0answers
141 views

Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
7
votes
3answers
2k views

What are affine spaces for?

I'm studying affine spaces but I can't understand what they are for. Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
7
votes
1answer
201 views

Two affine varieties are not isomorphic

Given the affine variety $A:=Z(y^{2}-P(x)) \subset \mathbb{C} ^{2} $, where $P(x)$ is a polynomial with $\deg P \geq 2$, I need to show that $A$ is not isomorphic to $ \mathbb{C}$. I know it ...
6
votes
1answer
7k views

What is the difference between linear and affine function

I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated
6
votes
2answers
553 views

Definition of Affine Independence in Brondsted's Convex Polytopes?

At one point in the book (An Introduction to Convex Polytopes, by Arne Brondsted) a definition of affine independence is given as follows, An n-family $(x_{1},...,x_{n})$ of points from ...
6
votes
4answers
652 views

Why is the affine hull of the unit circle $\mathbb R^2$?

In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $\mathbb R^n$ as $$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + ...
5
votes
3answers
370 views

The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?

Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) ...
5
votes
3answers
369 views

Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions

Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
5
votes
0answers
125 views

Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
4
votes
2answers
2k views

Decomposition of a nonsquare affine matrix

I have a $2\times 3$ affine matrix $$ M = \pmatrix{a &b &c\\ d &e &f} $$ which transforms a point $(x,y)$ into $x' = a x + by + c, y' = d x + e y + f$ Is there a way to ...
4
votes
1answer
82 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in ...
4
votes
1answer
149 views

Centre of a quadric

I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
4
votes
0answers
83 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
4
votes
0answers
87 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
4
votes
0answers
385 views

Is every convex-linear map an affine map?

Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. ...
3
votes
1answer
92 views

Prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$

I want to prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$, where the $p$'s and $q$'s are points. This is one of those ...
3
votes
4answers
138 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
3
votes
2answers
199 views

Polar form of (univariate) polynomials: looking for a proof

Recently I stumbled upon the following theorem — I'd like to read a comprehensible (i.e. understandable for an engineer) proof for it: Given a polynomial $F(t)$ of degree $n$, there exists a ...
3
votes
3answers
170 views

Closed subset of an affine variety… is it affine?

Preliminaries So, first of all let me give you the definitions I'm dealing with. Let $k$ be an algebraically closed field, and $\mathbb{A}^n = k^n$. An affine variety is a closed and irreducible ...
3
votes
1answer
424 views

Affine Independence $\iff$ Linearly Independent

I guess I'm having some trouble getting my head around the notion of affine independence. As I've been taught, a set of vectors $\{\vec{x_1},\ldots,\vec{x_n}\}\subset \mathbb{R}^d$ is affinely ...
3
votes
2answers
63 views

How to determine the period of a binary matrix?

As a part of my research project, I encounter a difficulty to compute the period of a binary cat matrix $C$, which is a square matrix of dimension $n$ with the following properties $|\rm{det}(C)|=1$ ...
3
votes
1answer
133 views

Complement of a line in $\mathbb{A}^2$ as an algebraic variety

I just started reading notes from an algebraic geometry course, and I'm curious about whether the complement of a line in $\mathbb{A}^2$ is always an algebraic variety. If so, what does it's ...
3
votes
2answers
37 views

Varieties strict inclusion infinitely long

Let $F$ be a field, and consider the affine space $F^n$. Could there be affine varieties $V_1,V_2,\ldots\subseteq F^n$ such that $V_1\supset V_2\supset\cdots$, and no two are equal?
3
votes
1answer
107 views

Prove, in this figure, that $EFGH$ is a parallelogram

In the following figure, $ABCD$ is a parallelogram, and $O$ is any point. Parallelograms $OAEB, OBFC, OCGD, ODHA$ are completed. Prove that $EFGH$ is a parallelogram. We can obtain a fairly trivial ...
3
votes
1answer
1k views

Difference between Projective Geometry and Affine Geometry

I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter I came across the following concepts.. Projective geometry is an extension of Euclidean ...
3
votes
1answer
241 views

Affine Spaces and Affine transformations

Can anyone please recommend a book that describes Affine Spaces and Affine Transformations? Many books i saw described it very briefly. Can anyone please suggest a book that deals with it in detail? ...
3
votes
1answer
217 views

Further questions on barycentric coordinates

Following on from my previous post... I'm going through this PDF file describing barycentric coordinates and trying to make sure I understand everything fully as I need to implement and support these ...
3
votes
1answer
39 views

Example: Irreducible component - affine varieties

Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
3
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1answer
105 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
3
votes
1answer
160 views

Fitting Shape in Circle for Shape Classification

I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
3
votes
1answer
121 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
3
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0answers
88 views

Smallest $\sigma$-algebra of $\Bbb A^n$ containing all affine algebraic subsets.

Let $k=\overline k$. What is the smallest $\sigma$-algebra $\Sigma$ containing all affine algebraic subsets? I am interested in the analogous question for $\operatorname{Spec} k[x_1,\dots,x_n]$, but ...
3
votes
0answers
59 views

Cardinality of quasiaffine variety

The excercise 1.4.8(a) of Hartshorne's Algebraic Geometry says Show that any variety of positive dimension over $k$ has the same cardinality as $k$. Using Hartshorne's notation, we define a ...
3
votes
1answer
196 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
3
votes
1answer
56 views

Set of all affine maps between two polytopes

Let $V$ be a finite-dimensional real vector space, let $P, Q \subseteq V$ be polytopes with $P \subseteq Q$. (Let a polytope be defined as the convex hull of finitely many points.) I'm interested in ...
3
votes
0answers
103 views

Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
3
votes
1answer
193 views

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
3
votes
0answers
441 views

Meaning of affine transformation

From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation. But in a book Multiple view geometry in computer vision by Hartley ...
2
votes
3answers
97 views

Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$

Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field. $\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane. Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$. I look at a line equation ...
2
votes
1answer
75 views

Whitney umbrella birational to $\mathbb{A}^2$ but not isomorphic

Define the Whitney umbrella as the affine surface $V(z^2 - yx^2) \subset \mathbb{A}^3$. I've come across an exercise that asks me to show that this surface is birational, but not isomorphic, to ...
2
votes
1answer
64 views

Affine space $A^n$ and definition of difference.

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is ...
2
votes
1answer
52 views

Skew planes in $\mathbb{A}^4$

Can there be two skew planes in $\mathbb{A}^4$? By this I mean two disjoint planes $\pi_1,\pi_2\subset\mathbb{A}^4$ such that their underlying direction vector spaces only intersect at zero.
2
votes
2answers
81 views

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$ So I've tried to put the equation of plane ...
2
votes
1answer
148 views

What does affinity means? (In categorical terms)

I know that traditionally, an affine space is "what is left from a vector space, after removing the origin". Given any set $X$ and a ring $K$, we can consider the set of all formal affine linear ...
2
votes
1answer
63 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
2
votes
1answer
83 views

How to find equation system describing affine space, having base of linear space and a vector

How to find equation system describing affine space, having base of linear 'overspace' and a vector? Suppose that I've vectors $\alpha$ and $\beta$, so that $W=\text{lin}(\alpha, \beta)$, and a ...
2
votes
1answer
38 views

Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$

I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$. On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the ...