# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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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 $i \in ... 1answer 76 views ### Isomorphisms of$\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 ...
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. ...
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 ...