# Questions tagged [projective-geometry]

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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\cdot$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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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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### 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $s_0\in S$. ...