Questions tagged [projective-geometry]

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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 ...
Marco Golla's user avatar
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2 votes
1 answer
131 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
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0 votes
1 answer
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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 ...
Learning math's user avatar
3 votes
2 answers
218 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
299 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
361 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
324 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
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5 votes
1 answer
337 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
488 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
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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 ...
Dac0's user avatar
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1 vote
1 answer
112 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
158 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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Centralness of planar convex regions - behavior under projective transformations

Background: For a planar convex region $C$ and an interior point $P$ we define: the centralness ratio at $P$ is $$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\...
Nandakumar R's user avatar
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12 votes
1 answer
1k 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
187 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
49 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
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2 votes
0 answers
110 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
62 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
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0 votes
0 answers
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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 ...
Display name's user avatar
1 vote
0 answers
59 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
102 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
139 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
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1 vote
0 answers
184 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
155 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
278 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
165 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
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2 votes
1 answer
224 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
273 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
148 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
519 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
402 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
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3 votes
2 answers
230 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
283 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
304 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
198 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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7 votes
0 answers
108 views

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 ...
Michael Rieck's user avatar
1 vote
0 answers
87 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 ...
Ming's user avatar
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3 votes
0 answers
34 views

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 ...
THC's user avatar
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5 votes
1 answer
272 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 ...
THC's user avatar
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2 votes
0 answers
63 views

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 ...
THC's user avatar
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2 votes
0 answers
54 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 ...
Dac0's user avatar
  • 285
3 votes
1 answer
192 views

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 ...
Puzzled's user avatar
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4 votes
0 answers
218 views

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 $n+1$ ...
bernardorim's user avatar
11 votes
3 answers
1k views

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 ...
Carlos Esparza's user avatar
4 votes
0 answers
123 views

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. ...
David Roberts's user avatar
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3 votes
0 answers
72 views

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 ...
Wlod AA's user avatar
  • 4,686
3 votes
3 answers
263 views

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 ...
Puzzled's user avatar
  • 8,832
4 votes
1 answer
196 views

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 ...
Malkoun's user avatar
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2 votes
1 answer
130 views

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 ...
gigi's user avatar
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3 votes
1 answer
2k views

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, ...
Đào Thanh Oai's user avatar

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