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Questions tagged [affine-geometry]

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.

11
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0answers
307 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 ...
6
votes
0answers
221 views

Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
6
votes
0answers
237 views

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

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
5
votes
0answers
111 views

Is the complement of a complex affine algebraic set in an irreducible complex affine algebraic set (path) connected in the euclidean topology?

Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is ...
5
votes
0answers
173 views

The Universal Property of Projective Space

Since the theories of affine and projective geometry are specified by certain axioms, we can consider the category $\mathcal{A}$ and $\mathcal{P}$ of affine and projective geometries, whose morphisms ...
5
votes
0answers
136 views

Most general space on which we can do calculus

I have two somewhat related questions: Question 1: What is the most general space (set of objects) on which we can do calculus? Is it a normed space, or can we relax the conditions a bit further? ...
5
votes
0answers
598 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 ...
5
votes
0answers
2k 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? $$ \begin{bmatrix}\cos\theta&-\...
4
votes
0answers
60 views

Translating and inflating a set of $k$-dimensional subspaces of $\mathbb F_p^n$ to form a cover by affine hyperplanes?

Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there ...
4
votes
0answers
109 views

Automorphisms of $\mathbb{A}^1_R$

When $R$ is an integral domain, the automorphisms of the affine line are all of the form $X \mapsto aX + b$ with $a \in R^\times$ and $b \in R$; the proof is the same as in the case of $R$ a field, ...
4
votes
0answers
140 views

Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
4
votes
0answers
126 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 ...
4
votes
0answers
2k views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
4
votes
0answers
143 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
votes
0answers
750 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 ...
4
votes
0answers
206 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
4
votes
0answers
122 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 ...
3
votes
0answers
100 views

Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
3
votes
0answers
114 views

What is the relationship between an affine space and an affine set?

I just learned what an affine space is and its basic properties. It is formally defined as $\Bbb A=(A,V,f)$ where $A$ is a set of points, $V$ is a vector space, and $f:A^2→V$ is a function st. for any ...
3
votes
0answers
189 views

Criterion for an affine isomorphism.

I am reading Don Taylor's book 'The Geometry of Classical Groups' and currently I am trying to understand the affine geometry section. There is a lemma which appears to be a criterion for a bijection ...
3
votes
0answers
168 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 ...
3
votes
0answers
169 views

Affine geometry book for physicist

I'm looking for a textbook to help me with understanding the geometry of Galilean relativity and the Galilean group. The reason is that I tried going through V.I. Arnold's Mathematical Methods, but ...
3
votes
0answers
298 views

Geometry book for the university with solved exercises (affine space, euclidean space, etc…)

I'm looking for a book with solved exercises of affine space, affine transformations, etc... I found a lot of books and pdf's with theory, but none of them contained solved exercises, and I'm having ...
3
votes
0answers
128 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf z}),\...
3
votes
0answers
168 views

Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
3
votes
0answers
3k views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
3
votes
0answers
125 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 quasi-...
3
votes
0answers
215 views

Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
3
votes
0answers
1k views

Meaning of affine transformation

From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation. But in a book Multiple view geometry in computer vision by Hartley ...
2
votes
0answers
43 views

Definition of a Product

For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx ...
2
votes
0answers
126 views

$GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $V$: $GL(V)$. For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic....
2
votes
0answers
98 views

Show that the affine transformation of a polyhedron is a polyhedron.

Let $P=\{x \in \mathbb{R}^n \mid Ax \geq b\}$ be a nonempty polyhedron for a matrix $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{m}$. Let $F:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be an ...
2
votes
0answers
44 views

Are all collinearity preserving bijections from $\mathbb R^3$ to itself of the form $Ax+b$ for $A \in M_{3\times3}(\mathbb R)$ and $b \in\mathbb R^3$?

Are all collinearity preserving bijections from $\newcommand{\RR}{\mathbb R}\RR^3$ to $\mathbb R^3$ of the form $Ax+b$ for $A$ in $M_{3\times3}(\RR)$ and $b \in \mathbb R^3$? In particular, let ...
2
votes
0answers
121 views

Understanding Constraint Matrix in Example Problem

I have found a pretty nice description of the polyhedra model in this paper. They are describing the math behind iterating through for-loops in a program (in last figure below), which forms a ...
2
votes
0answers
111 views

Finding the affine transformation from ordered set A to set B

I have an x-ray fluorescence detector which is used to measure the composition of different elements in a sample. The detector measures a signal which contains peaks at certain positions which ...
2
votes
0answers
83 views

Does an irreducible real affine algebraic set/ its complement has finitely many connected components in the Euclidean topology?

Let $n\ge 2$ and $V$ be an irreducible affine algebraic set in $\mathbb R^n$ . Then is it true that $V$ has only finitely many connected components in the Euclidean topology of $\mathbb R^n$ ? Does $\...
2
votes
0answers
90 views

$f \in \mathbb C[X,Y](=R)$ irreducible of degree $2$; then $R /(f) \cong R/(XY-1)$ or $R/(Y-X^2)$ as $\mathbb C$-algebras

For a field $k$ , we call a set $V \subseteq k^n$ an algebraic set if for some ideal $I $ of $k[X_1,...,X_n]$ , $V=Z(I)=\{(a_1,...,a_n)\in k^n : f(a_1,...,a_n)=0, \forall f \in I\}$ . Define $I(V):...
2
votes
0answers
68 views

When two affine subspaces are parallel?

Let $V_1$, $V_2$ be two projective subspaces of $\mathbb{P}^n$ such that $V_1\cap V_2 \neq \emptyset$, and let $\Pi$ be an hyperplane of $\mathbb{P}^n$. Is it true that the affine correspondence of $...
2
votes
0answers
57 views

coordinate ring of $X\times Y$

I was reading of Shaferevich's book of Basic Algebraic Geometry 1. On page 25 Example 4 it says that if $X$ and $Y$ are any closed sets in $\mathbb{A}^n$ and $\mathbb{A}^m$ respectively then $k[X\...
2
votes
0answers
368 views

Covariant derivative of contracted tensor product

I've followed a number of derivations of the coordinate expression for the covariant derivative of a one-form: $\nabla_\delta \omega_\alpha$. To summarize, one strategy is We know how to take ...
2
votes
0answers
48 views

Scaling in world space

In a hierarchical transformation system, where a node has one parent and children (Tree form) I want to scale an object with respect to world space axis. My transformation order is (Translate * Rotate ...
2
votes
0answers
44 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 ...
2
votes
0answers
174 views

Show that the singular locus $\Sigma$ of an affine variety $V$ contains no irreducible component of $V$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. Let $V=V_1\cup \cdots \cup V_r$ be a decomposition of variety into its irreducible components. Let $\Sigma$ be the singular ...
2
votes
0answers
912 views

What is the affine space and what is it for?

These two topics already exist: (preface: got in contact with affine space through computer graphics subject in university) What are affine spaces for? What are differences between affine space and ...
2
votes
0answers
123 views

Why affine variety not vector space variety?

I am new to algebraic geometry. A basic question baffles me: why is the setting the affine space not the vector space?
2
votes
0answers
91 views

Could we talk about affine spaces before vector spaces?

I was recommended the book Geometry by Michele Audin by a professor when I asked about learning more about affine geometry. I like the book, but it's raised a question. To me, it seems that it would ...
2
votes
0answers
128 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 ...
2
votes
0answers
53 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 ...
2
votes
0answers
39 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
votes
0answers
36 views

Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but isn'...