# Questions tagged [projective-geometry]

The projective-geometry tag has no usage guidance.

628
questions

2
votes

0
answers

79
views

### Group actions over non-closed fields

Let $G$ be an algebraic group over a field $k$ acting on a projective space $\mathbb{P}^n_k$. Assume that there exists a linear subspace $\Pi\subset\mathbb{P}^n_k$ of dimension $a$ such that for a ...

0
votes

1
answer

32
views

### Exceptional Lenz-Barlotti classes IVa.3 and IVb.3

On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...

0
votes

0
answers

24
views

### Extending homeomorphisms on closure spaces

Let $C$ be an infinite $T_1$ closure space, which is not a topological space. Suppose $C$ has the exchange property: for $x,y\in C$ and $A\subseteq C$
$$
\big( x\notin\overline{A}, \hspace{4mm} x\in \...

1
vote

1
answer

121
views

### Sum of two triangles in a projective plane modulo a conic

Given a conic $C$ in the complex projective plane, say $C=\{c:=x^2+y^2+z^2=0\}$, and two “triangles” (given as zeros of products of 3 linear forms $\ell=\{ax+by+cz\}$) $\ell_1\ell_2\ell_3$, $\ell'_1\...

1
vote

0
answers

31
views

### An algebraic characterization of dual translation projective planes

It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...

2
votes

1
answer

147
views

### Intersection in toric variety

In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension.
On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...

4
votes

1
answer

196
views

### Quotient of the plane by the standard Cremona involution

Consider the standard Cremona involution $i:\mathbb{P}^2\dashrightarrow \mathbb{P}^2$, $[x:y:z]\rightarrow [yz:xz:xy]$.
Let $Y$ be the blow-up of $\mathbb{P}^2$ in the three base points of $i$, so ...

1
vote

1
answer

68
views

### Uniqueness of a properly convex projective domain divisible by a group

Let us say that a discrete subgroup $\Gamma$ of ${\rm PGL}(n+1, \mathbb R)$ is convex divisible if there exists an invariant properly convex domain $\Omega$ of $\mathbb{RP}^n$, on which $\Gamma$ acts ...

6
votes

2
answers

237
views

### Embedding degree 1 Del Pezzo surfaces in $\mathbb{P}(1,1,2,3)$

In the projective bundle $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\oplus \mathcal{O})\rightarrow\mathbb{P}^1$ consider the hypersruface
$$
X := \{a_{00}y_0^2+a_{01}y_0y_1+a_{02}y_0y_2+a_{11}...

3
votes

0
answers

90
views

### Formulas for the line joining two points in the projective plane over a division algebra

Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane” $\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up ...

6
votes

0
answers

152
views

### What are the possible symmetry groups of n-point constructions in the projective plane?

Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.
I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel ...

3
votes

4
answers

539
views

### How big a class of lines can a non-linear transformation map to itself?

Edit: In the original version of this question, I wrote "lines through the origin" instead of "lines"; as Alexandre Eremenko points out in his answer, this makes the question too ...

3
votes

2
answers

318
views

### A paper of Borel (in German) on compact homogeneous Kähler manifolds

I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...

6
votes

0
answers

223
views

### A standard name for the algebraic structure on a projective line?

Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...

0
votes

0
answers

98
views

### Borromean rings on $\Bbb{RP}^2$ and octonions

If I draw a trefoil knot on a projective plane and draw a circle around it touching the three outer parts of the curve. I can view this as a division of the projective plane in 8 triangles, viewed as ...

3
votes

0
answers

89
views

### Are quadruples $abcd$ and $dcba$ always projectively equivalent in any projective plane?

It is well-known that for every line $L$ in a Pappian projective plane (i.e., a projective plane over a field) and any distinct points $a,b,c,d\in L$ the quadruples $(a,b,c,d)$ and $(d,c,b,a)$ are ...

7
votes

1
answer

266
views

### Computing $\pi_1$ of the complement of a non-singular plane curve

The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...

2
votes

1
answer

148
views

### A formula for the cross-ratio in terms of hyperbolic data

Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.
We will use the following convention for the cross-ratio $CR$ of ...

0
votes

1
answer

100
views

### Necessary and/or sufficient condition for invertibility of the gradient of a polynomial of $m$ variables, viewed as a self map of $\mathbb{R}^m?$

I was wondering whether the following is true, and if not, is something known in this direction?
Let $P:\mathbb{R}^m \to \mathbb{R}$ be a degree $r$ polynomial (not necessarily homogeneous) that ...

3
votes

2
answers

221
views

### Integer solutions to $x^2 + x + 1 = y^z$? [duplicate]

In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on ...

2
votes

2
answers

313
views

### A graphic representation of classical unitals on 28 points

I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...

9
votes

2
answers

376
views

### Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?

I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...

7
votes

1
answer

330
views

### A corollary of the affine Desargues axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...

5
votes

1
answer

361
views

### Fermat cubic hypersurfaces over finite fields

Consider the Fermat cubic
$$
X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}}
$$
over a finite field $\mathbb{F}_{q}$ with $q$ elements.
If $q \equiv 2 \mod 3$ then the projection $\...

10
votes

1
answer

506
views

### A projective plane in the Euclidean plane

Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...

2
votes

3
answers

347
views

### Moufang identities and Moufang plane

Moufang identities
$$x(y⋅xz)=(xy⋅x)z,$$
$$(zx⋅y)x=z(x⋅yx),$$
$$xy⋅zx=x(yz⋅x)$$
are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a ...

1
vote

1
answer

124
views

### Singularities of fibrations in conics

Consider a rank two vector bundle $E = \mathcal{O}(a)\oplus \mathcal{O}(b)\oplus \mathcal{O}(c)$ over $\mathbb{P}^1$. Fix coordinates $u_0,u_1$ on the base $\mathbb{P}^1$ and $v_0,v_1,v_2$ on the ...

1
vote

0
answers

164
views

### Cohomology of a stratified projective bundle

Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is ...

12
votes

1
answer

2k
views

### 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 ...

2
votes

2
answers

198
views

### 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 ...

1
vote

0
answers

54
views

### 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 $\mathrm{Gal}(\ell/k)$ ...

2
votes

0
answers

136
views

### 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}^...

1
vote

0
answers

63
views

### 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
votes

0
answers

89
views

### 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 ...

1
vote

0
answers

65
views

### 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 ...

1
vote

0
answers

112
views

### 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 ...

2
votes

1
answer

168
views

### 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 ...

1
vote

0
answers

230
views

### 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 ...

-1
votes

1
answer

196
views

### 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 ...

3
votes

1
answer

319
views

### 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
votes

0
answers

176
views

### 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 ...

2
votes

1
answer

230
views

### 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

274
views

### 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}(...

1
vote

1
answer

149
views

### 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}:...

11
votes

2
answers

531
views

### 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

438
views

### 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
answers

240
views

### 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

292
views

### 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
answer

330
views

### 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 ...

1
vote

0
answers

211
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 $s_0\in S$.
...