# Questions tagged [projective-geometry]

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601
questions

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### Mapping a cube to a sphere

I have been looking for a way to map a unit cube (with vertices $x^2=1$, $y^2=1$, $z^2=1$) to a unit sphere ($x^2+y^2+z^2=1$) with minimal distortion of the great circles formed by mapping the ...

4
votes

2
answers

138
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### Mori cones and projective morphisms

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely ...

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0
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42
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### Galois action on blow-ups related to field extensions of infinite degree

Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...

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0
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67
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### Computing the Pushforward for Arbitrary Coherent Sheaves in the Context of the Segre Embedding and K-Theory

Following thinking about a question from math overflow (and answering it https://math.stackexchange.com/a/4686391/299848) I was wondering about the topic:
Given the Segre embedding $\sigma: \mathbb{P}^...

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0
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54
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### Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?

Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on
$$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...

0
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0
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81
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### What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?

A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...

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49
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### Polytope of a projected toric variety

I was looking for such a result in the book by Cox, Little and Schenck but I'm not able to find a proper reference.
All of the following requirements are tacitly assumed to be in the projective ...

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0
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77
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### Linear span of tangential variety

Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...

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81
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### Linear system giving the projective embedding of the tangential variety

I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...

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79
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### Exterior product of Euler Exact Sequence

Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...

0
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1
answer

95
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### Definition of canonical pair

Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...

4
votes

1
answer

177
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### Normal bundle of a linear subspace

Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...

6
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142
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### Computing the automorphism scheme of projective space

$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...

4
votes

1
answer

196
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### Classification of quartic surfaces

Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...

2
votes

1
answer

264
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### Example showing that $\mathbb{P}^1$ does not preserve monics

Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(...

2
votes

1
answer

144
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### Geometry of contracted divisors

Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...

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votes

2
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463
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### Hypersurface of singular plane cubics

In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...

8
votes

1
answer

375
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### Automorphisms of projective spaces, and the Axiom of Choice

It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...

3
votes

2
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194
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### How do we define the type of a singularity on a cubic surface?

Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...

2
votes

1
answer

253
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### Is there a non-singular cubic surface that has a point where four lines intersect?

Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...

5
votes

1
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240
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### Which finite projective planes can have a symmetric incidence matrix?

As the title says. Which finite projective planes admit a symmetric incidence matrix?
I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always ...

2
votes

0
answers

158
views

### Semi-continuity of the Picard number

Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...

7
votes

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98
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### A spherical geometry claim related to the perspective 3-point problem

I have a simple claim in spherical geometry that has come out of my research into the so-called "perspective 3-point (pose) problem."
Here it is:
Fix three (distinct) great circles on the ...

1
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0
answers

80
views

### Lower bound of degree of ruled surface in $\mathbb P^n$

I have a question of Complex Algebraic surface in Beauville.
Let $S\subset\mathbb{P}^n$ be a (birationally) ruled surface of degree $d$ lying in no hyperplane.
Show that $d\geq 2 n-2$ if $S$ is not ...

3
votes

0
answers

25
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### Baer involutions fixing the same plane

Let $\mathbf{PG}(2,q^2)$ be the finite projective plane defined over the finite field $\mathbb{F}_{q^2}$. Then for each quadrangle, there is precisely one involution fixing it pointwise, and hence ...

5
votes

1
answer

248
views

### The map $k \mapsto \mathbf{PGL}_2(k)$

Consider the map $\zeta: \{ \mbox{division rings} \} \mapsto \{ \mbox{groups} \}: k \mapsto \mathbf{PGL}_2(k)$.
Is this map known to be an injection - in other words, if $k$ and $k'$ are nonisomorphic ...

2
votes

0
answers

50
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### Cross-ratio for projective lines over division rings

If one considers a projective line over a field $k$, then the cross-ratio $(w,x;y,z)$ is a well-known geometric tool.
But what if $k$ is not commutative, that is, if $k$ is a division ring ?
Is there ...

2
votes

0
answers

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views

### Classification of Moufang planes of real dimension 16

Incidence geometry is not really area of expertise so I'm asking here: are all Moufang planes of 16 dimension already classified?
I'm not just interested in the compact ones. Is there already a ...

4
votes

1
answer

174
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### Varieties connected by curves in projective spaces of small dimension

Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...

3
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0
answers

158
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### Quotient of $\mathbb P^n$ by the symmetric group $S_{n+1}$

The projective space ${\mathbb P}^n$ of dimension $n$ over a field (let's take $\mathbb C$ for simplicity) can be viewed as the space of homogeneous coordinates $[x_0:\cdots :x_n]$ in the $n+1$ ...

11
votes

3
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969
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### Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...

4
votes

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### Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation

It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...

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### Projective plane finite game

This is a 2-person game.
Let $\ P\ $ be any arbitrary projective space (of any dimension $\ \ge2$ and any cardinality, etc., but typically, let it be a finite plane over a field). Let $\ S_0\subseteq ...

4
votes

3
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### Varieties with few trisecant lines

Let $X\subset\mathbb{P}^N$ be an irreducible projective variety. Let's denote by $\mathcal{T}$ the following property: through a general point $x\in X$ there is no line intersecting $X$ in at least ...

4
votes

1
answer

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### Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$?

Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the ...

2
votes

1
answer

100
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### Class of the discriminant of a conic bundle

Let $X$ be a smooth projective variety and $E$ a vector bundle of rank $3$ over $X$. Moreover let $L \in Pic(X)$ be a line bundle and $$q:S^2E \rightarrow L$$ a $L-$valued quadratic form.
Then we can ...

3
votes

1
answer

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### Does this hexagon theorem have a name?

Question : Do you know this property of a hexagon?
Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ...

1
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0
answers

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### Matrix powers up to multiplicative factor

Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and
$$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$
I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \...

4
votes

0
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### Blowing-up a non reduced fiber

Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$.
I expect $\...

3
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0
answers

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### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...

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1
answer

66
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### Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$

Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...

2
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0
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### Nonlinear automorphisms of projective spaces and the axiom of choice

Let $k$ be a field and $\mathbf{P}$ a projective space over $k$. If we accept the axiom of choice (AC), then $\mathbf{P}$ has a basis and a dimension $m$, and if $m$ is finite, the automorphism group ...

2
votes

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### $0$-dimensional intersection in weighted projective space

Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by
$$
\phi(...

2
votes

1
answer

128
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### Surfaces with rational double points

Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...

4
votes

1
answer

151
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### Singularities of surfaces fibered in rational curves

Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point.
...

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154
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### Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...

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answer

406
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### Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines

The simplest case of the Fundamental Theorem of Projective Geometry states that, if $f: \mathbb{R}^2\to\mathbb{R}^2$ is a bijection that preserves lines – in the sense that if $L\subseteq\mathbb{R}^2$ ...

3
votes

0
answers

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### Anti-flag transitive affine planes

Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite.
Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...

2
votes

1
answer

154
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### Question regarding linear system of projective space

I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2....

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vote

1
answer

105
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### Number of orbits for abelian group actions

Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...