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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 ...
Robert B's user avatar
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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 ...
Taras Banakh's user avatar
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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 \...
Onur Oktay's user avatar
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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\...
Dima Pasechnik's user avatar
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 ...
Taras Banakh's user avatar
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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 $...
Robert B's user avatar
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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 ...
Robert B's user avatar
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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 ...
Roman's user avatar
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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}...
Robert B's user avatar
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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 ...
Gro-Tsen's user avatar
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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 ...
Gro-Tsen's user avatar
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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 ...
Steven Landsburg's user avatar
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 ...
Bobby-John Wilson's user avatar
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 $\cdot$ ...
Taras Banakh's user avatar
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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 ...
Maarten Havinga's user avatar
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 ...
Taras Banakh's user avatar
  • 41.1k
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 ...
Marco Golla's user avatar
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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 ...
Malkoun's user avatar
  • 5,118
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 ...
Learning math's user avatar
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 ...
Maarten Havinga's user avatar
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 ...
Taras Banakh's user avatar
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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 ...
Taras Banakh's user avatar
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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\...
Taras Banakh's user avatar
  • 41.1k
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 $\...
Puzzled's user avatar
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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 ...
Taras Banakh's user avatar
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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 ...
Dac0's user avatar
  • 295
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 ...
Puzzled's user avatar
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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 ...
IMeasy's user avatar
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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 ...
Harry van Langen's user avatar
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 ...
Puzzled's user avatar
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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 $\mathrm{Gal}(\ell/k)$ ...
THC's user avatar
  • 4,503
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}^...
Alon Yariv's user avatar
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 ...
Malkoun's user avatar
  • 5,118
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 ...
Display name's user avatar
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 ...
gigi's user avatar
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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 ...
gigi's user avatar
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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 ...
gigi's user avatar
  • 1,333
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 ...
BVquantization's user avatar
-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 ...
Puzzled's user avatar
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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 $H$ ...
Puzzled's user avatar
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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 ...
C.D.'s user avatar
  • 565
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\...
Puzzled's user avatar
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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}(...
Martin Brandenburg's user avatar
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}:...
Puzzled's user avatar
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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 ...
Puzzled's user avatar
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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 ...
THC's user avatar
  • 4,503
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 ...
mathlander's user avatar
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 ...
mathlander's user avatar
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 ...
Adelhart's user avatar
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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 $s_0\in S$. ...
Puzzled's user avatar
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