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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affine translation in direction of a vector

Suppose I have a line segment in 3D-space, having end-points $(a,b)$. I want to translate this segment by $w$ units in the direction specified by 3 angles $\alpha,\beta,\gamma$ with respect to ...
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1answer
77 views

The graph of a regular function is an algebraic set, and intersection of hypersurfaces is finite?

i have some problems with these exercises, can you give me a hint? Let $f:\mathbb A^n_k\rightarrow\mathbb A^m_k$ be a regular function. If $X\subset\mathbb A^n_k$ is an algebraic set, show that the ...
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93 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 ...
2
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1answer
292 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 ...
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0answers
124 views

Number of vector and affine subspaces of dimension $ k$ of $E$ over $\mathbb{F_q}$

Problem (comments after): Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$. Show that: ...
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1answer
206 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
2
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1answer
95 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 ...
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1answer
178 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
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0answers
79 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 ...
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1answer
109 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 ...
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1answer
166 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 ...
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0answers
108 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 ...
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1answer
98 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 ...
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1answer
73 views

who found that translation in N space is the same as shearing in N+1 space?

According to the wikipedia, Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective ...
3
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0answers
70 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 ...
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1answer
73 views

Geometric Deformations

There are geometric transformations such as translation, rotation and uniform scaling (Affine transformations). I am interested in knowing whether there is a separate class of transformations that ...
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0answers
53 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 ...
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0answers
114 views

Multi-affine function

Suppose I have a three-variable function $f(x_1, x_2, x_3)$, $f : \mathbb{R}^3 \to \mathbb{R}$. If it is linear for $x_1$, $x_2$ and $x_3$ we can say it has the form $f(x_1, x_2, x_3) = c_1x_1 + ...
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1answer
992 views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
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173 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 ...
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1answer
45 views

Affine transformation, if $L_1, L_2 - $ skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective

Could you tell me how to prove that if $f$ is affine transformation, $L_1, L_2 $ are skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective?
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1answer
188 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
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1answer
78 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 ...
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1answer
100 views

Why in the affine space can not we use Grassmann formula?

For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar. For this reason it is not worth the Grassmann ...
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0answers
87 views

Distance between two affine lines using determinant of Gramian matrix.

I've a task to find the distance in $E^4$ between: $L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$ and $M = [2,3,1,5] + \text{lin}((2,1,0,2))$ My efforts to find the correct solution: Let ...
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3answers
307 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 ...
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0answers
52 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 ...
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3answers
313 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?
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0answers
74 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.
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1answer
188 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 ...
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1answer
33 views

affine variety of infinitely many polynomials can be represented as an affine variety of its finite subset

Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is ...
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1answer
58 views

Invariants of point sets in an affine space

A distance between a pair of points in an affine space is invariant under translation, rotation and reflection. An angle in a triangle whose corners are tree points is also invariant under scaling. ...
2
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1answer
127 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 ...
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1answer
93 views

Intersection of affine subspaces of finite codimension in Hilbert space

I'm wondering whether the following assertion is true: Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
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1answer
152 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 $q+1$ ...
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1answer
241 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 ...
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0answers
93 views

Prove something is affine?

For any subspace $K$ and any point $u$, prove $K+u$ is affine. Or if you have an affine set $V$ and point $u$, then prove $V-u$ is a subspace.
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1answer
74 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.
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2answers
90 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 ...
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4answers
96 views

Find the equation of plane containing line described by

Please help me in this really easy task Find the equation of plane containing line described by $x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
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0answers
66 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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2answers
173 views

Equation of plane — point/vector pedagogy

Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
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1answer
70 views

Variety Affine Space

I have the following question which i'm not sure how to work out... $For\ f=6x^2y-xy^2-2y^3+1\ and\ \ h=3x-2y\ \in \mathbb{C}[x,y]$ Show that V(f,h) is empty. What can you say about the ideal ...
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3answers
561 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 $\mathbb{R}$, ...
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4answers
174 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
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1answer
703 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 ...
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1answer
360 views

Intersection of affine subspaces is affine

So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in ...
5
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3answers
427 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) ...
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3answers
220 views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the ...
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2answers
72 views

Affine geometry about parallelogram

If ABCD is a parallelogram and M,N,P,Q are points on it sides then MNPQ is a paralellogram iff the diagonals intersect at a common point (i.e the diagonals of MNPQ and ABCD intersect at the same ...