4
votes
0answers
66 views

Incidence variety fo Grassmmanians

Let $k$ be an algebraic closed field (say, $\text{char}(k)\neq 2$), $n \in \mathbb N\setminus \{0\}$ and $G(m, n) = G(m, \mathbb P^n(k))$ the variety of Grassmmanian of $m$-dimensional linear ...
0
votes
1answer
27 views

Möbius transforms with a common fixed point

Let $f,g$ be two Möbius transformations with a common fixed point $z_0$. Show that the Möbius transformation $f \circ g \circ f^{-1} \circ g^{-1} $ is either parabolic or the identity. Möbius ...
1
vote
1answer
36 views

Tangent lines of conics

Let $k$ be algebraically closed. Let $P\in k[x,y,z]$ be a homogeneous quadratic polynomial. Let $C$ be the zero locus of $(P)$ in $\mathbb{P}^2$. Let $Q \in \mathbb{P}^2$. Is there a tangent line at ...
5
votes
1answer
53 views

Problem about parallel curve- differential geometry

Let $\alpha (s)$ , $s\in [0,L]$, be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve $\beta(s)=\alpha (s) +\lambda n(s)$, where $\lambda$ is a positive ...
2
votes
1answer
26 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 ...
0
votes
1answer
33 views

factor to find an algebraic expression for the length and width of the rectangle

the area of the rectangle is defined by $$ 6x^2+13x-28 $$ so far, i have decomposed the expression to get $$(3x-4)(2x+7)$$ but, now i need to find the length and width and that's where I have a bit of ...
1
vote
0answers
22 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
4
votes
2answers
56 views

Degree 2 Fermat curve

I'm trying to solve the following exercise: Prove that the variety $V\subset \mathbb{CP}^2$ defined by $x^2+y^2+z^2=0$ is isomorphic to $\mathbb{CP}^1$. What I've done: I tried to define an explicit ...
5
votes
1answer
77 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
0
votes
1answer
55 views

Proof: The coordinates of the witch of Agnesi curve

I need to prove that the coordinates ofthe witch of Agnesi curve is: $$x=2a\cot \theta$$ and $$y=2a\sin ^2 \theta$$ Any idea how to prove it? And I don't understand how we got $a$... (because the ...
0
votes
0answers
33 views

Basic on free morphism

Let $X$ be a smooth variety of dimension $n$ over an algebraically closed field of characteristic zero, and let $f: \mathbb{P}^{1} \rightarrow X$ be a nonconstant morphism. Then $$f^{\ast}T_{X} \cong ...
3
votes
2answers
81 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
2
votes
2answers
87 views

Quotient of a local ring at a point is a finite dimensional vector space

$f,g\in \mathbb{C}[x,y]$ are irreducible polynomials, the varieties $V_1=V(f)$ and $V_2=V(g)$ are not equal. Is the ring $\mathcal{O}_p/(f,g)$ a finite dimensional vector space over $\mathbb{C}$? ...
1
vote
1answer
32 views

If two points of intersection of a cubic and a line have real coordinates then so does the third.

Let $F\in \mathbb{R}[X,Y,Z]$ be a homogeneous cubic and let $C=V(F)\subset \mathbb{P}_{\mathbb{C}} ^2$. Let $P,Q\in C$ and let $L$ be the line through $P$ and $Q$. Suppose that $R$ is the third point ...
1
vote
0answers
40 views

Pascal's theorem by Bezout's theorem

I need to prove the following theorem Let the hexagon $ABCDEF$ be inscribed in the nondegenerate conic $q=V(f)$. Assume that $A,B,C,D,E,F$ are distinct. Let $P=\overline{FA}\cap \overline{CD}, ...
0
votes
0answers
32 views

Show that the following are algebraic sets :

$\{(t,t^2,t^3) \in \mathbb A^3(K) \ \ | \ \ t\in K \}$ $\{( (cos(t),sin(t) ) \in \mathbb A^2(\mathbb R) \ \ | \ \ t \in \mathbb R \}$ The set fo points in $\mathbb A^2(\mathbb R)$ whose polar ...
3
votes
1answer
68 views

$Z\subset \mathbb{P}^n$ irreducible iff its pre-image in $\mathbb{A}^{n+1}-\{0\}$ irreducible

I'm having trouble with this question (it's a homework question). If $p:\mathbb{A}^{n+1}-\{0\} \rightarrow \mathbb{P}^n$ is the canonical projection and $Z\subset \mathbb{P}^n$ is closed, then $Z$ is ...
2
votes
1answer
72 views

Exercise 2.3 from Hartshorne's algebraic Geometry.

2.3) A scheme $(X,\mathcal{O}_X)$ is reduced if for every open set $U\subset X$, the ring $\mathcal{O}_(U)$ has no nilpotent element. b) Let $(X,\mathcal{O}_X)$ be a scheme. Let ...
5
votes
1answer
61 views

A simple sheaf computation

I am currently taking my first course in algebraic geometry and am stuck at te following problem, which I am sure is simple. Consider $Y := \mathbb P^1 \times \{x\}$ as a closed subscheme of $\mathbb ...
8
votes
3answers
155 views

show the set of points $(t^3, t^4, t^5)$ is closed in $A^{3}$

Show the set of points $X = \{(t^3, t^4, t^5) \}$ with $t\in k$ is closed in $A^{3}=k^3$ and find three generators of $\mathcal{I}(X)$. This is a homework question, so please don't provide full ...
2
votes
1answer
44 views

Prove that if $V$ and $W$ are affine varieties, then $V \times W$ is an affine variety.

I am working on the problem from "Ideas, Varieties and Algorithms" by David Cox, John Little and Donal O'Shea. Here is the homework problem for my course. Let $V \subset k^n$ and $W \subset k^m$ ...
0
votes
0answers
39 views

Show that the map defines an isomorphism

I have been struggling with this exercise for a while. Let $f\in k[V]$ where $V$ is a variety in $\mathbb{A}^{n}(k)$ and let $Gr(f)=\{(a_{1},....a_{n},a_{n+1})\in ...
2
votes
1answer
53 views

check my work on Sharfarevich section 3 problem 4

The problem is to find the irreducible components of the affine variety $X=Z(y^2=xz, z^2=y^3)$ over an algebraically closed field $\mathbb{k}$ and show that each component is birational to ...
1
vote
1answer
72 views

Shifting to a disjoint subvariety

Let $X\subset \mathbb{A}^8$ be a subvariety of dimension 3. Show that there is a vector $v\in k^8$ such that $(X+v)\cap X=\emptyset$. (remark: $X+v=\{x+v|x\in X\}$)
0
votes
1answer
58 views

Symmetries of a square and it's similarity to the Division Algorithm

I need help with this question: (each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion. This question is ...
1
vote
1answer
25 views

Ring of rational functions ideal generators

There is an affine variety $X\subset \mathbb{A}^n$ with its ring of rational functions which is the quotient ring of $\mathbb{k}[X]$ (each $f\in \mathbb{k}(X)$ has a form $\frac{p}{q}$ where $q$ does ...
0
votes
0answers
28 views

Smooth conics in linear subspaces of $S^2U^*$

Let $U$ be some $3$-dimensional vector space over some field $\mathbb{k}$. It is possible to consider the projective space $\mathbb{P}(S^2U^*)$ as a space of conics on the projective plane ...
3
votes
2answers
114 views

Prove that a complex valued polynomial over two variables has infinitely many zeroes

This is a homework question that I am struggling with. Given a polynomial over the complex numbers in two variables, show that the polynomial has infinitely many zeroes. So let's say that the ...
1
vote
1answer
42 views

Projection of a hypersurface is dominant

Let $F\in \mathbb{k}[x_1,...x_n]$ be a polynomial and let $V(F)=X$ be the set of its zeros in $\mathbb{A}^n$. We take some point $p\in \mathbb{A}^n\setminus V(f)$. I have just understood what the ...
2
votes
1answer
53 views

Vector bundle on a quadric $Q$

The problem: Consider the smooth quadric $Q=V(X_{0}X_{1}+X_{2}X_{3}+X_{4}^{2})\subset\mathbb{P}^{4}$ and the line $L=V(X_{0},X_{2},X_{4})$ contained in it. Prove that there exists a vector bundle $F$ ...
2
votes
0answers
44 views

Vector bundle on a quadric [duplicate]

The problem: Consider the smooth quadric $Q=V(X_{0}X_{1}+X_{2}X_{3}+X_{4}^{2})\subset\mathbb{P}^{4}$ and the line $L=V(X_{0},X_{2},X_{3})$ contained in it. Prove that there exists a vector bundle $F$ ...
3
votes
1answer
68 views

Divisors on a complex torus

I'm asked to prove the following fact: On a complex torus $X$ every canonical divisor is principal and vice-versa. At this moment I know only the basic properties of divisors and that, if $K$ is a ...
2
votes
1answer
44 views

Decomposing an affine algebraic set into irreducible ones

Let $X\subset\mathbb C^4$ be given by the system \begin{align} x_1x_4 - x_2x_3 &=0\\ x_1x_3 - x_2^2 &=0 \end{align} I need to decompose this into a union of irreducible sets. The obvious ...
2
votes
0answers
84 views

Unique factorization in 3-sphere coordinate ring

For $n\geqslant 1$ define $$A_n=\mathbb{C}[X_0,X_1,\dots,X_n]\Bigg/\left(\sum_{i=0}^{n}X^2_i-1\right).$$ I would like to prove that $A_3$ is a unique factorization domain. For $A_2$ it is not true ...
3
votes
0answers
94 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
1
vote
0answers
27 views

Help on this example of gluing varieties

Let $V_1=\{(x_1,y_1) \in \mathbb{C}^2\mid y_1^2=x_1^2-3x_1+2\}$ and $V_2=\{(x_2,y_2) \in \mathbb{C}^2\mid y_2^2=2x_2^2-3x_2+1\}$. We set $U_i=\{(x_i,y_i) \mid x_i\neq 0\} \subset V_i,\ i\in\{1,2\}$. ...
1
vote
1answer
49 views

Affine algebraic curve is Riemann surface

The problem: Let $P\in\mathbb{C}[z]$ be a non-constant polynomial with simple zeros. Show that the affine algebraic curve $X=\{(z,w)\in\mathbb{C}^2\,:\,p(z)=w^2\}$ is a (connected) Riemann surface. ...
1
vote
1answer
45 views

Proving that maximal ideals in coordinate ring determine unique point

I have the following homework question: Let $X \subseteq \mathbb{A}^n$ be an irreducible algebraic subset, and let $\mathbb{K}$ be algebraically closed. Show that every maximal ideal in ...
4
votes
1answer
73 views

Finding the Vanishing Set of an Algebraic Set

We've been given the set $X = \{(t^3,t^4,t^5) \in \mathbb{A}^3 \mid t \in \mathbb{A}^1\}$ (where the underlying field $\mathbb{K}$ is infinite), and have been asked to show that $X = \mathbb{V}(J)$ ...
-1
votes
1answer
31 views

Density Calculations Help needed Please

At -189 °C argon freezes to form a crystalline solid with a face-centred cubic lattice. The shortest distance between the centres of two adjacent argon atoms is 3.82 Å. The length of the unit cell ...
2
votes
1answer
81 views

Constructing a rational map from a divisor

This problem arose in an algebraic geometry course I'm taking, and my understanding of it comes from Shavarevich's "Basic Algebraic Geometry." The question is this: Given a projective variety $$X = ...
3
votes
1answer
107 views

Rational functions on $V(xw-yz)$

Problem: Let $k$ be an algebraically closed field and $V=V(xw-yz)=\{(x,y,z,w)\in\mathbb{A}^4(k): xw-yz=0\}$. Let $\Gamma(V)$ be the ring of coordinates of $V$ and $k(V)$ its field of fractions. Let ...
3
votes
1answer
110 views

Proving the algebraic set corresponding to a polynomial is infinite.

This is an exercise from Miles Reid, Undergraduate Algebraic Geometry. The proof has two parts, one of which I can do and one of which I can't. I could have some misunderstandings about notations here ...
3
votes
1answer
111 views

quasi-affine/projective varieties | f=g on dense subset | diagonal subset | how to show that (f,g) is continuous?

Let $f: X \to Y$ and $g: X \to Y$ be morphisms in the category $(QProj-k)$ (its objects are quasi-projective and quasi-affine $k$-varieties). Show that $f=g$ if and only if $f$ and $g$ are identical ...
3
votes
0answers
54 views

Picard group of $G(k, n)$ saying about automorphisms

Let $G(k, n)$ be the Grassmannian of $k$-dimensional subspaces of $K^{n}$, $K$ a field, embedded in $\mathbb{P}^{N}$ by the Plücker embedding. In Harris' Algebraic Geometry, A First Course, Theorem ...
10
votes
1answer
373 views

Atiyah-MacDonald help with exercise 5.10

This is an exercise from Atiyah-MacDonald, if someone can give an idea on how to prove that $a)\Rightarrow b)$: Let $f:A\rightarrow B$ a ring homomorphism. a) ...
2
votes
0answers
41 views

Being a morphism of quasiprojective varieties is a local property

Let $X,Y$ be quasiprojective varieties and $\phi \colon X \to Y$ be a map. Suppose there exists a cover $\mathscr U = \{U_i\}_{i \in I}$ of open subsets $U_i \subset X$ such that for every $i \in ...
4
votes
1answer
76 views

Isomorphisms of $\mathbb P^1$

Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form $$ \phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1) $$ where $\begin{pmatrix} a & b ...
3
votes
1answer
94 views

Morphisms between quasiprojective varieties preserve irreducibility

Let $X,Y$ be two quasiprojective varieties and $\phi \colon X \to Y$ a surjective morphism. Let $Z \subset Y$ a closed set such that $\phi^{-1}(Z)$ is irreducible. Prove that $Z$ is irreducible. ...
3
votes
1answer
150 views

Cremona transformations are birational maps

Consider the following map, which is a Cremona transformation: $$ \begin{split} f\colon & \mathbb P^2 \dashrightarrow \mathbb P^2 \\ & (x:y:z) \mapsto (xy: xz: yz) \end{split} $$ I have to ...