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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1answer
25 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 ...
2
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0answers
34 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
41 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 ...
0
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1answer
23 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
18 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 $X=V(\langle x^5-x^4+2x^2y-y^2\rangle) \subset \mathbb{A}^2(\mathbb{C})$
0
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1answer
12 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
8 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: $$ ...
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1answer
19 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
10 views

Affine spaces points

I got a question of affine spaces and the question is about prove given L is an affine space. $(L=U+A)$ where $U$ is a vector space and $A$ is a point i.e. $(2,3)$ . So my question is to prove this ...
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0answers
24 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
11 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 ...
0
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1answer
29 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
18 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 ...
0
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1answer
31 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
26 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
0
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1answer
20 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
58 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 + ...
1
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1answer
45 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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0answers
9 views

Lenght of the affine transformation $s_{\varphi_n, \, 1} \cdot \dots \cdot s_{\varphi_1, \, 1}$

Let $\{\varphi_1, \, \dots , \, \varphi_n\}$ be a subset of positive roots of a root system $\varPhi$ and consider the affine reflections $\{s_{\varphi_n, \, 1}, \, \dots , \, s_{\varphi_1, \, 1} \}$ ...
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2answers
36 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
43 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 ...
0
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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
46 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 ...
8
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4answers
184 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
46 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
27 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 ...
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0answers
14 views

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
37 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
25 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 ...
0
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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 ...
2
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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 ...
0
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1answer
37 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
68 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 ...
1
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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 : ...
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1answer
37 views

What's the intuition behind the definition of the tangent space of $\Bbb R^2$?

I'm reading a book on differential forms and on page one it defines the tangent space to $\Bbb R^n$. In what follows I've translated the statements into two dimensions for simplicity. Let $p$ be a ...
2
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0answers
33 views

affine transformations, strategy for finding invariant straight lines

At first lets introduce some notation. $\mathcal{A}^n$ is a $n-$dimensional affine space and $V$ is its associated vector space. For any affine subspace of $\mathcal{M}$, its associated vector space ...
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0answers
7 views

Transforming a vector configuration to an affine point configuration

I am reading the first chapter of Oriented Matroids by Bjorner et. al, in which they consider the set of vectors $\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_6$ given as the columns of the following ...
0
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1answer
29 views

Affine variety of geometric progressions.

Let $M$ be affine variety $M \subset \mathbb A^n$. $z\in M \Leftrightarrow z = (x, xy, xy^2, ... , xy^{n-1}),where\hspace{2mm} x,y \in \mathbb C, x \neq 0$. I need to find the equations $f_1, ..., ...
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3answers
89 views

How to gain an intuition of the affine function's definition?

Here is the definition of Affine Functions according to Stephan Boyd (EE263 Stanford) : 1- I believe linearity is more restrictive property of a function than being affine since it requires $f(0) = ...
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1answer
26 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
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0answers
37 views

Why is half space not affine?

I've read that half spaces are convex but not affine. I'm trying to understand this geometrically. Does it mean that if I connect any 2 points in the half space, it may result in a line that extends ...
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0answers
8 views

There are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \cap l_{3} $ is fixed point.

$l_{1}, l_{2}, l_{3} $ are 3 pairwise orthogonal lines in $\mathbb{E_3}$ Prove that there are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} ...