# Tagged Questions

Questions about projective space in geometry, a space which can be seen as the set of lines through the origin in some vector space. As such it is a special case of a Grassmanian. See https://en.wikipedia.org/wiki/Projective_space

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### Euler characteristic of the projective plane (using embedding diagram)

Make the square into the projective plane $\mathbb{P}$ by identifying edges and compute the Euler characteristic by embedding the following graph onto the surface: Here is my diagram of the ...
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### Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
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### Is $\mathbb{R}P^1 \times \mathbb{R}P^1 = \mathbb{R}P^2$? If it isn't what does it look like?

We know that $\mathbb{R}P^1$ may be conceptualized as the set of all lines through the origin in $\mathbb{R}^2$. Alternatively, it may be conceptualized as a hemisphere of $S^1$ with antipodal points ...
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### The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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### If two projective lines both intersect four given projective lines, must the two lines be parallel on an affine open?

L.S., I am trying to solve an exercise of my algebraic geometry course, which is as follows. Given four projective lines in $\mathbb{P}^3$, show that the number of lines intersecting them all is ...
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### What properties single out $\operatorname{Spec}(\mathbb{k})$-schemes that are quasi-projective varieties over $\mathbb{k}$?

I have a question in algebraic geometry that I would like to ask: Let $\mathbb{k}$ be an algebraically closed field. Is there a property $P$, phrased in the language of schemes, such that ...
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### Understanding problem 2.6 in Hartshornes algebraic geometry book .

Can anybody help me in understanding the hint given in the problem $2.6$ in Chapter $1$ of Hartshorne's Algebraic Geometry book ? I cannot see why $A(Y_i)$ can be identified with degree zero ...
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### How to write the real projective plane as a pushout of a disk and the mobius strip?

I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout? Also, how can I ...
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### pencil of cubic curve passing six points

Let [1,0,0],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3] be a six points in general position. The question is how can determine the pencil of cubic curve passing through these points? Many thanks.
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### projective nullstellensatz proof

I have to give a talk about projective varieties, including the projective nullstellensatz. As I'm not really into algebra or algebraic geometry, I've got some problems with the proof. Projectiv ...
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### Group acting on a Projective Space

Let $G$ be an algebraic (zariski closed) subgroup of $SL(n,C)$ for some algebraically closed field $C$. Now $G$ acts on an $n$-dimensional vector space $V$ over $C$ where $V$ is a solution space of a ...
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### Projection on the subspace perpendicular to a vector

I am running an algorithm and during one of the steps of the algorithm, I have to update a matrix $B$ but projecting every column of the matrix, $B_j$, j=1,...,k, on the subspace perpendicular to a ...
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### Lines on a Quintic Threefold

We work over an algebraically closed field $k$. I've been given the exercise of showing (using only the technology introduced in the first chapter of Shafarevich's Basic Algebraic Geometry) that ...
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### Rings of Regular functions, and regular maps between Quasi Affine to Quasi Proj. Varieties.

I have studied classical algebraic geometry a while ago. I want to sum up in short as possible everything regarding their rings of regular functions. If my understanding not correct, please correct me....
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### Show that any two distinct lines in $\Bbb P^2$ intersect in one point.
Show that any two distinct lines in $\Bbb P^2$ intersect in one point. Proof(My attempt). Let $L_1, L_2$ be any two distinct lines in $P^2$. Write $L_i = V (a_iX + b_iY + c_iZ), i = 1,2$. It ...