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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2
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3answers
98 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
32 views

Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
2
votes
1answer
23 views

Are the quasi-affine subsets of $\mathbb{A}^1_F$ necessarily open or closed?

For what follows, my definition of a quasi-affine subset of $\mathbb{A}^n_F$ is one which can be written as $Z_1\setminus Z_2$, where $Z_1$ and $Z_2$ are closed subsets of $\mathbb{A}^n_F$. (I think ...
2
votes
2answers
27 views

If we delete two points $x,y$ from $\mathbb{A}^1$, can we without loss of generality assume $x=0, y=1$?

My intuition is that we can assume this. More precisely, what I mean is, suppose $\mathbb{A}^1_k$ is the affine space over an algebraically closed field $k$. If $x,y$ are any two distinct points in ...
2
votes
1answer
40 views

Is a similarity map necessarily affine linear?

My text on fractal geometry introduces the following definition: A map $S: \mathbb R^n \to \mathbb R^n$ is called a similarity map if $$\exists c>0 \ \forall x,y \in \mathbb R^n: ...
2
votes
1answer
51 views

Computing the Zariski cotangent space

I'm an extreme beginner with algebraic geometry and am trying to get used to things. Say I have some (algebraically closed) field $k$, in $k^2$ I want to compute the Zariski cotangent space, let's say ...
2
votes
1answer
93 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
92 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
67 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
89 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
199 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
35 views

Domain of definition of $(1-y)/x$ on $x^2+y^2=1$?

I'm self-teaching myself some basic algebraic geometry, and I wanted to double check something that seems too easy. An exercise sheet I found asks to compute the domain of definition of the rational ...
2
votes
1answer
53 views

Why is $f(X)$ open or closed if $f:X\to\mathbb{A}^1(k)$ is regular?

I have a question about a certain property of regular maps into $\mathbb{A}^1(k)$. This is my notation for the affine space over $k$, algebraically closed. Suppose $f:X\to \mathbb{A}^n(k)$ is a ...
2
votes
1answer
39 views

Difference between Euclidean space and $\mathbb R^3$

What is the difference between Euclidean space and $\mathbb R^3$? I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is ...
2
votes
1answer
59 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
2
votes
1answer
81 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
274 views

Best book to learn Affine Geometry?

I'm going to learn Affine plane as well as affine Geometry. Unfortunately, my text book (not in English) is not good at all, so please recommend some book you think it's good for self-learning (and ...
2
votes
1answer
122 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
39 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 ...
2
votes
1answer
107 views

Projective geometry well defined bijection

I consider the sphere $\mathbb S^n:=\{x\in\mathbb R^{n+1}: \|x\|=1 \}$ and the equivalence relation $x\sim y:\Leftrightarrow x=\pm y$. How can it be shown that the inclusion $\mathbb ...
2
votes
1answer
103 views

Equation of the line in an affine plane over a polynomial field

What are some examples of this? Say for $F_{4}$. I know this is a very simple question, but I can't find any info on it. Edit: Yes, I was thinking of $F_{2}[x]/(x^2+x+1)$. I was confused.
2
votes
1answer
36 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
2
votes
1answer
38 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
2
votes
1answer
90 views

Affine hull of the intersection of two convex sets

Is it true that the affine hull of the intersection of two convex sets is the intersection of the affine hulls of these sets? Where the intersection of the two convex sets is non empty? Many thanks!
2
votes
1answer
77 views

Does convexity of all projections imply convexity in higher dimensions?

If I have $n$ 2-dimensional convex "regions" that are the projections along $n$ (independent) dimensions of a n-dimensional compact subspace: Does that imply that the latter is convex? Is the ...
2
votes
1answer
173 views

On affine spaces, distances, angles, and coordinates

Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that $\forall \, a \in ...
2
votes
1answer
453 views

Applications of the fundamental theorems of affine and projective geometry.

The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a (semi-) affine/projective ...
2
votes
1answer
97 views

non-affine functions

it is obvious that if $f$ is an affine function, then $f$ has this property: there exist two function $g$ and $h$ such that $f(t+s)=g(t)+h(s)$ for all $t,s \in\mathbb{R}$. My question is: is there ...
2
votes
1answer
109 views

Finding a coset

I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the ...
2
votes
1answer
363 views

Turning affine planes into projective planes

How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$? Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point ...
2
votes
1answer
89 views

Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the ...
2
votes
1answer
53 views

Decomposition of shear matrix into rotation & scaling

How can I decompose the affine transformation: $$ \begin{bmatrix}1&\text{shear}_x\\\text{shear}_y&1\end{bmatrix}$$ into rotation and scaling primitives? $$ ...
2
votes
1answer
23 views

How to orthogonalize a set of 2x2 matrices?

I have set of 2D affine transformations of images and I need to modify the transformations such way that they become as close to rotations as possible to minimize distortions of images. Let the ...
2
votes
0answers
28 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$. How is it possible to deduce that $G$ is conjugated to a subgroup of $GL_2(k)$ ?
2
votes
0answers
71 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
2
votes
0answers
38 views

A question for epigraph and affine function

I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction. First, I quickly recall some definitions: A (closed) halfspace is a ...
2
votes
1answer
37 views

Find any affine transformation that swaps affine lines

The task is to find any affine transformation that will swap the following two lines: $$L_1:(1,1,1) + span((1,0,2))$$ $$L_2:(1,0,1) + span((1,0,-1))$$ From what I understand there is a number of ...
2
votes
0answers
51 views

How to describe the variety for lines through the origin

Let $I(x)$ be an ideal generated by the affine variety $X$. $I(X)=\langle xy,yz,zx\rangle$ gives the three coordinate axes through the origin (because it is the intersection of the $3$ planes $xy=0$, ...
2
votes
0answers
37 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
2
votes
0answers
34 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
2
votes
0answers
258 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
2
votes
2answers
106 views

Prove that $\text{(BE)}\|\text{(JF)}$ using vectors.

Problem Let $\text{ABC}$ be a triangle and let $\text{I}$ , $\text{J}$ and $\text{K}$ be points such that: $\vec{\text{BI}}=\frac{1}{2}\vec{\text{IC}}$, $\vec{\text{AJ}}=2\vec{\text{JB}}$ and ...
2
votes
0answers
162 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
2
votes
0answers
73 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
2
votes
0answers
52 views

$\varphi$ affine if with $V = \{V_{i}\}_{i=1, .., m}$ affine covering such that $\varphi^{-1}(V_{i})$ affine

I found an exercise of mine, that I solved, but now I am not sure anymore about the details, and some parts of what I wanted to say there. I have the affine open sets $V_{1}, ..., V_{m}$ and ...
2
votes
1answer
179 views

Prove that for every point in one-sheeted hyperboloid, there exists at least one line which is full contained in it

Please help me with the task: Prove that for every point in one-sheeted hyperboloid, there exist at least one line, which is full contained in it. Firstly, I've noticed that I can transform the ...
2
votes
0answers
51 views

$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$

Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$ Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis. I don't ...
2
votes
0answers
68 views

Affine set and linear equation

Prove or disprove the following statement. For any affine set C in R^n, there exists a solution set of linear equation that express C.
2
votes
1answer
161 views

Affine Transformation ― correct direction of scale

due to the fact that I am not mathematician I hope the question wont be ejected cause of triviality. But here we go: what is given: In an svg graphic, I have an element on which several are ...
2
votes
1answer
201 views

Finding the singularities of affine and projective varieties

I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed. I'm not sure if the definition I've been given is ...