0
votes
1answer
37 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
2
votes
1answer
39 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
43 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 ...
3
votes
1answer
53 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
0
votes
2answers
36 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...
4
votes
2answers
94 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
1
vote
1answer
77 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...
2
votes
1answer
31 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 ...
3
votes
1answer
50 views

Example: Irreducible component - affine varieties

Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
2
votes
1answer
69 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 ...
1
vote
1answer
39 views

Do all the solutions have to be in an affine variety?

An affine variety $V(X)$ is the zero-locus of a set of polynomials. So if the variety is generated by the polynomial $y-x=0$ in $\mathbb{R}^2$, then do all the solutions (i.e., every point satisfying ...
2
votes
0answers
49 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 $xy=0$, ...
3
votes
2answers
38 views

Varieties strict inclusion infinitely long

Let $F$ be a field, and consider the affine space $F^n$. Could there be affine varieties $V_1,V_2,\ldots\subseteq F^n$ such that $V_1\supset V_2\supset\cdots$, and no two are equal?
2
votes
0answers
209 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
95 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 ...
1
vote
1answer
63 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 ...
4
votes
0answers
89 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
votes
1answer
190 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 ...
0
votes
0answers
92 views

Intersection of two affine varieties

I'm trying to determine the points where two affine varieties defined over $\mathbb{R}$ intersect. Obviously I can just plot them and see what they look like, but I'd have to just look and guess where ...
2
votes
1answer
83 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
0answers
56 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 ...
3
votes
1answer
95 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 ...
3
votes
1answer
145 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 ...
3
votes
0answers
93 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 ...
3
votes
0answers
66 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 ...
2
votes
0answers
48 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 ...
8
votes
0answers
145 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 ...
3
votes
3answers
211 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 ...
10
votes
3answers
275 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?
7
votes
1answer
206 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 ...
5
votes
3answers
389 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) ...
0
votes
3answers
165 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 ...
1
vote
1answer
97 views

Detecting the type of singularity with the Jacobian

Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then ...
2
votes
1answer
186 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 ...
10
votes
3answers
2k views

$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
10
votes
2answers
525 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
0
votes
1answer
53 views

finding two polynomials that their roots are a given line.

Given a field $F$ and $A = F^3$. we define $L$ to be the line that goes through the points: $(8,1,-1)$, $(5,0,-1)$. My object is to find two polynomials $q(X_1,X_2,X_3)$, $p(X_1,X_2,X_3)$ in ...