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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2answers
174 views

An irreducible quadric hypersurface is rational?

Here quadric hypersurface just means it is generated by a polynomial with degree 2. I can guess the idea is to project the hypersurface from a fixed point P, to some plane by drawing a line through P....
-2
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1answer
33 views

Affine hull of two points in R4 [closed]

I try to describe an affine hull of two points (1,3,2,4) and (1,4,2,3) so i try to make the linear equation which describe it .
2
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0answers
46 views

Affine geometry textbook

What's a good recommendation for a book on affine geometry at the undergrad level? I ask because I skimmed through the first bit of Vladimir Arnold's Mathematical Methods of Classical Mechanics and ...
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0answers
21 views

The points of an affine variety where a given rational map is regular

This is a very general question, because I can't seem to find this in the notes for the course. So let $X$ be an affine algebraic variety and $K[X] = K[\mathbb{A}^n]/I(X) = K[\overline{x_1},...,\...
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1answer
34 views

Let $B=A+t\vec{AC}$ with $t\ne1$, show that $A=B+s\vec{BC}$ for $s=t/(t-1)$

I have to prove the following: Let $B=A+t\vec{AC}$. Let $t:=(A, B, C)=\frac{\vec{AB}}{\vec{AC}}$. Prove that $(B, A, C)=\frac{t}{t-1}$. I've been trying by two different ways but I always obtain ...
0
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1answer
41 views

Every irreducible component of an affine cone contains its vertex

Let $X=V(F_1,...,F_k)\subset \mathbb{P}^n$with $F_i\in k[X_0,...,X_n]$ an projective algebraic set. Let $C(X)\in \mathbb{A}^{n+1}$ the affine cone over $X$, that is $C(X)=\theta^{-1}(X)\cup \{(0,...,...
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0answers
29 views

Example of affine varieties with a restriction on the dimension of an irreducible component of their intersection

On Hartshorne's book of algebraic geometry, exercise 2.11 (c) page 13 it's ask to prove that for any two linear varieties $Y,Z$ in $P^n$, with $dim Y=r$, $dim Z=s$, if $r+s-n\geq 0$, then $Y\cap Z\neq ...
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2answers
80 views

Show that halfspace is not affine.

Let us define half-space as $$ C = \{x\mid a^Tx\leq b\} $$ Intuitively (or geometrically), I understand why halfspace is not affine. But while I prove that half-space is convex, it seems to hold for ...
0
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1answer
46 views

Why is this function an affine function?

Why is the following function an affine function? $$f(x)=(P^{1/2}x,c^Tx)$$ I learnt that affine functions have the pattern like $f(x)=Ax+b$, is there any relation between the two function? Maybe I ...
3
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2answers
130 views

Any continuous function with the mean value property is affine

A function $f(t)$ on an interval $I=(a,b)$ has the mean value property if $$f\left(\frac{s+t}{2}\right)=\frac{f(s)+f(t)}{2} \quad s,t\in I$$ Show that any affine function $f(t)=At+B$ has the mean ...
0
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1answer
26 views

Finding an invertible element that is not invertible in a subring of a coordinate ring

First, let $K$ be an algebraically closed field. Now let $\phi: X \rightarrow Y$ be a dominant map of affine varieties, so namely $\phi^\star: K[Y]\rightarrow K[X]$ is an injection, and so we have a $...
0
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1answer
43 views

Is it possible to define the vertex coordinates of a dodecahedron as integer multiples of basis vectors in an affine space?

I am trying to think of an affine space defined by cells (whose edges may differ in length), such that the coordinates of all the vertices in a regular dodecahedron can be expressed as integer ...
0
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1answer
46 views

Ideals of affine variety

Let $X=V(y^2+x^2y-x^2)$ be an affine variety of affine space of 2 variables. What is the ideals of affine variety, $I(X)$. We know that $X$ consists of the curve $y^2-x^2y=x^2$. So how do we determine ...
3
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1answer
70 views

basic question on varieties (algebraic geometry)

I study basic algebraic geometry and I saw this exercise: V is the complement of the twisted cubic in $$ A_c^3. $$ i.e. $$ V = A_c^3 - \{(t^3, t^4, t^5) \mid t\in c\}. $$ 1. How can I proove that V is ...
4
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2answers
47 views

Non-isomorphic two-variable varieties in characteristic 2

Let $K$ be an algebraically closed field of characteristic $2$, and we will consider affine varieties of $\mathbb{A}^2$. Let $X = Z(y-x^2)$ and $Y = Z(xy-1)$. I have shown through some exhausting case ...
1
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1answer
116 views

Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
4
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1answer
69 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u \...
2
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1answer
49 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P \...
0
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1answer
302 views

Decompose affine transformation (including shear in x and y)

How can I fully decompose an affine transformation matrix that includes tx, ty, rotation (theta), scale-x (sx), scale-y (sy), shear-x and shear-y? Using this matrix as example: $$A = \begin{pmatrix}...
2
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1answer
61 views

Cohomology of sheaves of abelian groups on affine space

Is it true that $$\mathrm{H}^p_\mathrm{Zar}(\mathbb{A}^n_\mathbb{K}, \mathcal{F})=0$$ for any $p>1$ and $\mathcal{F}$ (non constant) sheaf of abelian groups? If not, is it true for some fields/...
1
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0answers
33 views

Explanation of $\ker (\bar{m}-\bar{id})^2 \cap \{x_0=1\}$

Let $m$ be an isometry on $\mathbb{R}^2$ which is a composition of a reflection and a translation. The way to find the axis of the isomtry is by solving: $$\ker (\bar{m}-\bar{id})^2 \cap \{x_0=1\}$$ ...
1
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1answer
50 views

Proving $\gcd(f_i)=1\Rightarrow \mathbb{A}_\mathbb{C}^n\setminus \{f_i\}$ is not affine

I need to prove the following lemma: Lemma: Let $f_i\in \mathbb{C}[x_1,\dots,x_m]$ s.t. $\gcd(f_1,f_2,\dots,f_n)=1\quad(1<n\le m)$. Prove that the variety $V=\mathbb{A}_\mathbb{C}^m\setminus\{...
3
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1answer
38 views

Orthogonality in affine subspaces

(Note: edited after the first comment) Let $A, B$ be nonempty subspaces of an affine space $E$ such that $A \cap B = \emptyset$. I am asked to prove that there are $a \in A$ and $b \in B$ such that $...
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1answer
33 views

Showing space closed under affine combinations is translation of vector space

I'm struggling to reconcile two different definitions of an affine space. The definition in my course notes is: An affine space in $\mathbb{R}^n$ is a non-empty subset closed under affine ...
1
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1answer
32 views

why should add one column using Moore-Penrose pseudoinverse

I have a code from someone that I dont understand: This code is written in matlab and the function is to estimate linear geometric transformation [1] of a matrix using pinv. The size of first matrix ...
1
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1answer
183 views

The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} \...
1
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1answer
74 views

Why does isomorphism follow from the natural bijection of Hom sets

If X and Y are varieties, and Y is affine, there is a natural bijective mapping of sets $$\operatorname{Hom}(X,Y)\xrightarrow{\sim}\operatorname{Hom}(A(Y),\mathscr O(X))$$ where the left are ...
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2answers
28 views

If $W$ admits an injection of $k$-algebras in its coordinate ring, then $W$ is an unirational variety

I'm studying algebraic geometry from "Introduction to algebraic geometry" by Hassett, and I did not understand a step in his proof of the following result (page 52): "If $W$ is an affine variety ...
3
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1answer
150 views

Affine varieties over finite fields

I read in this paper (http://www.math.iitb.ac.in/~srg/preprints/Chandigarh.pdf) that the following set is an affine variety: $V_f=\{(t_0,...,t_N)\in \mathbb{F}_p^{N+1} : f(t_0,...,t_N)=0 \}$ where $f$...
0
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1answer
50 views

affine 3D transformation reconstruction

How can we get the affine 3D matrix in case we have the 3D rotation matrix, the 3D translation vector, the scale factors and the shearing factors? A = SHEARING (4,4) * ScaleMatrix (4,4) * ...
0
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1answer
21 views

Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if and only if $\phi$ is invertible in $\mathbb{C}[V]$.

This is an exercises in Ideals, Varieties and Algorithms by Cox et al. Let $V\subset \mathbb{C}^n$ be a nonempty variety. Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if ...
4
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0answers
102 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
1
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1answer
33 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
4
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0answers
185 views

Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
10
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2answers
629 views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
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1answer
61 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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1answer
135 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
1
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1answer
23 views

Question about affine isometric action

Recently, I read the book Kazhdan's Property (T). There is a lemma on the page 75 (Lemma 2.2.1) as following: Lemma. Let $\pi$ be an orthogonal representation of $G$ on $H^0$. For a mapping $\alpha: ...
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1answer
47 views

Intersection of a cone and a plane

In $\mathbb{R}^3$, given the cone $K$ and the plane $E_c$ with the equations $4x^2=y^2+z^2$ and $z=c(1-x)$. How do I find out which different geometric objects I get for all $c\geq 0$ if I intersect ...
2
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0answers
47 views

Automorphism of $\mathbb{A}^2$ which maps the finite set of points to the finite set of points

Let $\mathrm{k}$ be infinite field. $P_1,\dots,P_n, Q_1,\dots,Q_n \in \mathbb{A}^2$ and $P_i \neq P_j, Q_i \neq Q_j$. I want to find automorphism(in a.g. sense) which maps $P_i$ to $Q_i$. I have tried ...
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1answer
35 views

How to prove, that $\sim$ is an equivalence relation? (affine equivalence)

Two quadrics $Q_1$ and $Q_2$ in $\mathbb{R}^n$ are affine equivalent, $Q_1\sim Q_2$, if there exists an affine map $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with $Q_2=f(Q_1)$. How do I prove, that $\...
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2answers
232 views

affine vs projective tranformation

I'm trying to grasp the difference between the affine and projective transformations...I got the point of the line at infinity but their matrix representation is not yet clear enough: ...
2
votes
1answer
112 views

Affine subspace of $\Bbb{R}^{27}$

I have affine subspace $K$, $K \subset \mathbb R^{27}$. It's elements are solutions of system of linear equations $Ax=b, b \in R^{16}$. What are maximum and minimum dimensions of said subspace, if I ...
0
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1answer
89 views

Equation of the affine transformation that fixates a certain line

I have to find the equation of the affine transformation of the affine plane $A_2$ that (1) fixates the line $s: x + y - 1 = 0$ and (2) such that $A(Q)=P$, where $Q(1,2)$ and $P(2,1)$. How should I ...
3
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1answer
53 views

How can affine coordinate rings be canonically identified as $k$-algebras?

Exercise 1.5 of Hartshorne asks us to show (in one direction) that any affine coordinate ring $k[x_1,\dots,x_n]/I(Y)$ is a finitely-generated $k$-algebra with no nilpotents. The second part is quite ...
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2answers
324 views

“Averaging” transformation matrices?

I have a question on how best to "average" transformation matrices. Say that I have n number of 4x4 transformation matrices, and I wanted to find a matrix that approximated each one of the n 4x4 ...
2
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0answers
36 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
2
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1answer
76 views

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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0answers
40 views

Two meanings to affine independence? (help me clear up my misunderstanding)

I must be misunderstanding something. Let's look at the following two definitions for a set of points $S=\{v_1,v_2,...,v_k\}$ to be affinely independent: 1) S is affinely independent if the set $\{...
6
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1answer
239 views

Intersection of affine varieties is affine

Let $M,N\subset\mathbb{P}^n$ quasiprojective varieties such that there exist isomorphisms $i\colon M\rightarrow Z(a)\subset \mathbb{A}^m$ and $j\colon N\rightarrow Z(b)\subset \mathbb{A}^m$ for ideals ...