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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Isomorphisms and orthogonality in projective space.

I've been studying projective space in algebraic geometry for a few days from Perrin's 'Algebraic Geometry: An introduction'. In the first page of chapter III it reads ...projective space $P^n$ ...
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3answers
30 views

How many convexly independent vectors there are in $\mathbb{R}^n$

I know there are n linearly independent and n + 1 affinely independent vectors in $\mathbb{R}^n$. But how many convexly independent there are? I think there are infinity number of them because if I ...
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2answers
50 views

Give an example of an affine space that is not a vector space

We know that any vector space is an affine space, but can you give an example of an affine space which is not a vector space? I don't know any such examples. This is an interview question, not ...
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1answer
39 views

Book(s) about Affine geometry.

A quick look on Stack Exchange enabled me to discover "Geometry" from Michele Audin which is very close from what I'm expecting but there isn't the correction of the exercices. To be more specific, I'...
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0answers
21 views

How to create a polygon on an affine space?

If I had an affine space with a basis $\{a_0, a_1, a_2\}$, I could use these points to either create a triangle or select other three points $\{c_0, c_1, c_2\}$ on the plane to create the triangle $\...
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1answer
51 views

Zariski closure of $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}\subseteq \mathbb{C}^4$?

Let $V= \mathcal{V}(\langle xy-zw\rangle)\subseteq \mathbb{C}^4$ be an affine variety. The set $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}$ is a torus contained in $V$. I am trying to ...
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0answers
4 views

finite partitions of the square that separate all equipotent sets of points

This question asked whether there exists a finite partition of $[0, 1]^2$ and a finite set of points in $[0, 1]^2$ that can't be affinely transformed to fall into one part of the partition. I would ...
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1answer
93 views

A combinatorial question

Let us look on a $p\times p$ board (the $(\mathbb{F}_p)^2$ plane) with a single piece on the down left corner $(0,0)$. This is a special piece that has $3$ legal moves: Moving one step up $\pmod p$ ...
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0answers
61 views

fitting points into partitions of a square

A friend of mine came up with the following problem: Let $\{X_1, X_2, ..., X_n\}$ be an arbitrary finite partition of the unit square $[0, 1]^2$. Let $\{P_1, P_2, ..., P_m\}$ be a finite set of ...
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0answers
22 views

Show that $E / \sim $ is an affine space over $V/U$

Let $E$ be an affine space over the vector space $V$, and let $U \subseteq V$ be a vector subspace. We define the equivalence relation $$P \sim Q := \exists v \in U \text{ such that } P = Q + v$$ on $...
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1answer
29 views

Dimension of product of affine varieties

Let $X\subset \mathbb{A}^n_K$ and $Y\subset \mathbb{A}^m_K$ be affine varieties. How can I prove that dimension of the product variety $X\times Y \subset \mathbb{A}^{m+n}_K$ is dim$X$+dim$Y$? Here I ...
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0answers
35 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} & ,...
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2answers
43 views

Finding modular inverse of every number mod 26?

I am looking at cryptography, and need to find the inverse of every possible number mod 26. Is there a fast way of this, or am i headed to the algorithm every time?
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0answers
12 views

Affinities that transform lines in other lines

In $\mathbb C^2$ I have the following three lines: $r_1:3x-y+3=0, r_2:y=0, r_3:x-i=0$ I want to find all the affine transformations such that $f(r_1)=r_2, f(r_2)=r_3, f(r_3)=r_1$ I was thinking ...
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1answer
24 views

Closest point on Affine Line to Origin?

I have had two courses of linear algebra and was always aware that affine spaces were a thing. What I know about affine spaces are that they are similar to vector spaces but do not necessarily have to ...
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0answers
16 views

Affine transformations in the complex

In $\mathbb C^2$ I have the following three lines: $r_1:3x-y+3=0, r_2:y=0, r_3:x-i=0$ I want to find all the affine transformations such that $f(r_1)=r_2, f(r_2)=r_3, f(r_3)=r_1$ How can I do it? ...
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0answers
22 views

Examples of applications of the Theorems of Pappus and Ménélaüs.

I'm going to present an exposition about applications of the Theorems of Pappus and Ménélaüs. I need some simple examples of these two theorems. Any links, please?
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1answer
14 views

Does this map define a rational map?

$\phi(x,y)=\frac{y-x^2}{x^2}$ for $\phi:X\to\mathbb{A}^1(\mathbb{C})$ $X$ being a variety $X=V(\langle x^5-x^4+2x^2y-y^2\rangle) \subset \mathbb{A}^2(\mathbb{C})$
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1answer
13 views

Find an affine linear map given two vectors

Find an affine linear map $$\mathbb{Z}_2^5\to\mathbb{Z}_2^5$$ that sends $(0,1,0,0,1)$ to $(1,0,0,1,0)$. So I know that an affine linear map is one of the form $Az+b$ where $b,z\in\mathbb{Z}_2^5$ ...
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0answers
9 views

Image and preimage of projection onto affine space

I saw this task some time ago, but I still cannot do it without horrible many calculations. I will be grateful for any hint. Affine subspace $H$ of $\mathbb{R}^4$ given is by equations: $$ \begin{...
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1answer
21 views

Affine contractions from linear contractions?

Let $V$ be a linear space. Consider a contractive linear map $M:V\mapsto V$, $$ \|Mv\|\leq \|v\| \quad \text{for all vectors } v\in V. $$ Now, for some fixed vector $c\in V$, the question is to sort ...
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0answers
31 views

Prove this is an Affine Space

I have a poor knowledge about affine spaces but I know there are ways to prove an affine space using closed affine combinations theory. But this equestion is different. Which is; Let $L = \{(x,y)\in \...
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0answers
13 views

Order of parabolic subgroups of affine Weyl groups

I have a question about computing the order of an arbitrary parabolic subgroup of an affine Weyl group $W_a$. Given a proper subset $I \subset S_a$ associated with the reflections for the fundamental ...
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1answer
30 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert &...
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1answer
19 views

Intersection of two circles in projective space

I have checked the existing question Intersection of two circles. and model for intersection of two circles in the complex projective plane - I do not think either of these answers my question. The ...
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1answer
33 views

Affine Subspace Proof

Question: Suppose that $V$ is an $m$-dimensional affine subspace of $\mathbb R^n$, with $m < n$. show that there exist linearly independent vectors $a_1, \dots, a_{n-m}$, and scalars $b_1, \dots ,...
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0answers
27 views

Show that in finite affine geometry all lines are on the same number of points step by step proof.

i'm wondering if someone could please give me a step by step proof of the following problem Show that in finite affine geometry all lines are on the same number of points
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1answer
23 views

Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
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1answer
59 views

Every line in $\mathbb{R}^2$ can be described…

I came across such a statement: Let $A = \mathbb{R}^2$, $a,b,c \in A$ be points (we treat $\mathbb{R}^2$ as an affine space). Then any line $L \in A$ can be described as $$L = \lbrace s_1 a + ...
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1answer
47 views

affine variety/space vs. toric variety

I think I'm not quite clear on the meaning of a toric variety... Could someone explain the relation/difference between the affine variety/space and the toric variety? I know that affine ...
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2answers
37 views

Is the affine curve $y^2=x^4+y^4$ in $\mathbb A^2$ singular?

Is the affine curve $y^2=x^4+y^4$ in $\mathbb A^2$ singular or nonsingular? Find the singularities and show the types of the singularities if the curve is singular. Let $f(x,y)=x^4+y^4-y^2$. $...
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0answers
12 views

Affine application

Consider the affine space $\mathbb{C}^2$ on the field $\mathbb{C}$. How to show that the function $\mathbb{C}^2 \rightarrow \mathbb{C}^2 : (x,y) \mapsto (\bar x, \bar y)$ transforms lines into lines ...
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1answer
46 views

When does an affine manifold inherit a (quotient) group action?

By an affine manifold I mean a real $n$-dimensional manifold $M$ with charts whose transition functions are in the affine group $Aff(\Bbb R^n)$. There are several other equivalent definitions https://...
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1answer
11 views

Affine subspaces proposition

I try to prove that a subset of an affine space is an affine space iff it contains the line through every two distinct points of it.
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0answers
10 views

Number of points of an affin line

I want to show that every affin line of $A(V)$ , where $V$ is a vector space on $K$ and $A(V)$ its affin space, has $|K|$ points
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2answers
25 views

Show that some set is an algebraic set

Show that $\{(t,t^2,t^3):t\in k\}$ for a field $k$ is an algebraic set. Just by looking at the points, i see that they are zeros of $F_1(x,y,z)=xy-z$, $F_2(x,y,z)=xz-y^2$ and $F_3(x,y,z)=yz-x^5$. I ...
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2answers
48 views

Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or $\...
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4answers
186 views

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
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2answers
60 views

The composition of two homotheties is either a homothety or translation

Let $f$ and $g$ be two homotheties that don't have the same center. What kind of affine transformation is their transformation? I know it can be either a homothety or translation, but I don't know ...
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1answer
28 views

Affine variety as subset of another affine variety

I am trying to understand why the following statement is true: If $S$ and $S'$ are subsets of $\mathbb{K}[X_1,...,X_n]$ such that $S\subseteq S'$, then $\mathcal{V}(S') \subseteq \mathcal{V}(S)$....
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0answers
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Existence of affine hull of set S

"Obviously, the intersection of an arbitrary collection of affine sets is again affine. Therefore, given any $S \subset R^{n}$ there exists a unique smallest affine set containing $S$ (namely, the ...
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1answer
41 views

How to prove: Every affine set can be expressed as the solution set of a system of linear equations [closed]

How can I prove that every affine set can be expressed as the solution set of a system of linear equations? Please note that set $C \subseteq \textbf{R}^n$ is said to be affine if for any $x_1 , x_2 ...
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1answer
27 views

Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
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1answer
35 views

Check the following sets are algebraic or not?

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
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2answers
109 views

Finding the defining equations for a simple quotient variety

First of all let me note that I have no experience at all with modern algebraic geometry so if at all possible I would appreciate an answer not involving the concept of a scheme. I have however some ...
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1answer
38 views

Flexible affine variety

Let X be an affine variety. Aut(X) is automorphisms group of X. SAut(X) is a subgroup spanned by all unipotent(image of additive group of field K) subgruops. X is called flexible if SAut(X) act ...
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0answers
16 views

How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
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5answers
69 views

Prove 3 vectors are collinear

I am asked to prove A(2,4), B(8,6), C(11,7) are collinear using vectors. I can work AB by subtracting A from B and BC by subtracting B from C in vector form. I can say that BC = 2AB. But I don't ...
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1answer
37 views

Show that a set is an affine subset

Let $V$ be a vector space over $\mathbb{F}$ and $S \subseteq V$ a nonempty set. Theorem: $S$ is an affine subset of $V$ if and only if $$ \forall u,v \in S, \forall \lambda \in \mathbb{F}, \...
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0answers
17 views

Dimesion of an affine variety- solution verification

I have affine variety $V=\{(t,t^2,t^3)|t\in\mathbb{Q}\}$ and if I'm right I have $I(V)=(y-x^2,z-x^3)$ and coordinate ring is $\mathbb{Q}[V]\cong \mathbb{Q[t]}$ (I set $x=t$). I have this definiton : ...