# Tagged Questions

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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### 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....
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### 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 .
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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/...
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### 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: $$\ker (\bar{m}-\bar{id})^2 \cap \{x_0=1\}$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f$...
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### 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) * ...
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### 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 ...
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### 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 ...
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### 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 $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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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### “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 ...
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### 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 ...
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 $V$ ...