Questions tagged [projective-geometry]

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Kaehler analogue of very ample line bundle

In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a ...
Pierre Dubois's user avatar
10 votes
1 answer
283 views

Freiman inequality for projective space?

This question is suggested by some results in a paper I am writing. I would like to write it down there but want to make sure that it is not known or at least MO-hard. Freiman's inequality states ...
Hailong Dao's user avatar
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1 vote
0 answers
104 views

Completion of infinite projective space

I would like to ask for a reference regarding the completion of infinite-dimensional Projective Space (both Real and Complex). Since in the infinite-dimensional projective space you can take sequences ...
userm's user avatar
  • 11
2 votes
1 answer
172 views

Non-commutative projective lines

There have been many approaches to the notion of projective line: combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...
THC's user avatar
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4 votes
2 answers
602 views

Principled construction of the quaternions

Is there a construction of the quaternions that doesn't proceed through generators and relations, and which makes the connection with 3D rotations clear? I'm not happy with Clifford Algebra as an ...
wlad's user avatar
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2 votes
0 answers
152 views

Terminal and log canonical singularities

Let $D$ be a divisor with at most terminal singularities in a smooth projective variety $X$. Is the pair $(X,D)$ log canonical?
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0 votes
1 answer
215 views

Rank of matrices and secant varieties

Consider the Segre embedding $\mathbb{P}^n\times \mathbb{P}^n\rightarrow \mathbb{P}^N$, and let $S\subset\mathbb{P}^N$ be its image. Then $rank(Z)\leq k$ implies that $Z\in Sec_k(S)$. Moreover if $Z\...
user avatar
4 votes
2 answers
340 views

Maps between grassmannians with inclusion property

Edit: According to the comment of L. Spice I changed the inclusion sign to the subset sign. Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What ...
Ali Taghavi's user avatar
3 votes
0 answers
125 views

$\left< 15\right>^7/15$-womcode construction

In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
Alexey Ustinov's user avatar
1 vote
1 answer
157 views

Collineations of projective spaces and isomorphisms of fields

For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for ...
Taras Banakh's user avatar
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0 answers
45 views

How to projectivize an infinite-dimensional $L^{2}$ space and do fourier analysis on said projectivization

I'm doing work with repelling (that is, non-contracting) linear operators on hilbert spaces, and I wondered if it might be worth my while to study my operators on a projectivization of my hilbert ...
MCS's user avatar
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12 votes
1 answer
305 views

Geodesic preserving diffeomorphisms of constant curvature spaces

Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$. I would like to have a classification of all diffeomorphisms $X\to X$ which map ...
asv's user avatar
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3 votes
0 answers
104 views

Where can I learn about the discrete symmetries of the complex projective plane (or space)?

I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
guest78's user avatar
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7 votes
1 answer
462 views

The projective functor $\mathbb{P}^n: \operatorname{CRing} \to \operatorname{Set}$ is not representable: categorical argument

Using a "geometrical" argument of dimension, like the one here, one can show that the projective space is not affine. I am interested in showing that, but using a categorical argument, i.e. I want ...
sagirot's user avatar
  • 455
2 votes
1 answer
423 views

Join of two intersecting varieties

Suppose I have two smooth projective varieties $X$ and $Y$ in $\mathbb{P}^n$, that intersect along a smooth subvariety $Z$. Is there a formula to compute the degree of the join variety $J(X,Y)$ of $X$ ...
IMeasy's user avatar
  • 3,707
3 votes
0 answers
86 views

Canonical sheaf of Schubert cycles

Suppose we have a smooth subvariety $X\subset Gr(2,n)$ of a Grassmannian, that can be expressed as usual as a linear combination of Schubert cycles. I would like to obtain information on the canonical ...
IMeasy's user avatar
  • 3,707
2 votes
0 answers
240 views

Direct sums of invertible sheaves commuting with global sections and the functor of points approach

I am looking at the Stacks Project's treatment of the functor of points for projective space. Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
Luke's user avatar
  • 413
1 vote
0 answers
152 views

What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?

Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line. What is the ...
windsheaf's user avatar
  • 435
0 votes
1 answer
176 views

What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?

Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? ...
Malkoun's user avatar
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7 votes
1 answer
139 views

Combinatorial curves in combinatorial projective planes

Suppose $\mathcal{P}$ is an axiomatic projective plane (a nondegenerate point-line incidence geometry in which there is a unique line on any two different points, and a unique intersection point for ...
THC's user avatar
  • 4,313
3 votes
1 answer
228 views

Relation between the decomposition invariants of a projective reduced curve and its normalization

Let $X$ be a reduced projective scheme over $k$ which is of pure dimension 1. Let $\pi: X \to \mathbb{P}_k^1$ be a finite (hence affine, surjective and flat) morphism of schemes having degree $n$. ...
windsheaf's user avatar
  • 435
2 votes
0 answers
441 views

Morphisms of smooth varieties

Let $f:X\rightarrow Y$ be a surjective morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that $Y$ is smooth projective and all the fibers are ...
user avatar
1 vote
0 answers
125 views

Morphisms whose reduction is projective

IIUC Remark 5.3.5 in EGA II says that there exist proper non-projective morphisms $X\rightarrow Y$ where $Y$ is the spectrum of a finite-dimensional $\mathbb{C}$-algebra such that the induced morphism ...
user avatar
3 votes
1 answer
153 views

First-order logic of projective planes over fields [closed]

Suppose $\mathbb{P}^2(k)$ is a projective plane defined/coordinatized over a commutative field $k$. Is the first-order logic of the plane completely determined by the first-order logic of $k$ ? (In ...
THC's user avatar
  • 4,313
2 votes
0 answers
361 views

Intersection of two quadrics in $\mathbb{R} P^5$

Is there an ''algorithmic'' way to get that intersection of two quadrics $$x_1 y_1 -x_2 y_2 - z_1^2+z_2^2=0$$ and $$x_2 y_1 + x_1 y_2 -2z_1z_2=0$$ inside $\mathbb{R}P^5[x_1:x_2:y_1:y_2:z_1:z_2]$ is ...
Filip's user avatar
  • 1,617
0 votes
1 answer
38 views

Gluing simplices through a common point/ realisation of a convex simplicial polytope

Given $m≥d+1$ a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that 1) they all share the common vertex M 2) the ...
giulio bullsaver's user avatar
4 votes
1 answer
382 views

Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
Squid with Black Bean Sauce's user avatar
7 votes
2 answers
2k views

Embeddings of flag manifolds

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
user avatar
3 votes
1 answer
410 views

Automorphisms of singular hypersurfaces

Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$. A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...
user avatar
2 votes
1 answer
1k views

Cohomology of tangent sheaf of a hypersurface

Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
user avatar
2 votes
0 answers
139 views

Cubic 3-fold singular along a curve

Does there exists a cubic or quartic $3$-fold $X\subset\mathbb{P}^4$ such that $Sing(X)$ is a smooth curve $C$ of genus $g(C)\geq 2$ and $X$ has $A_1$-singularities along $C$?
user avatar
3 votes
0 answers
151 views

Reference request: invariants/tableaux functions for 4 lines in $P^3$

Does anybody have a reference for invariants of configurations of linear subspaces in the projective space? In particular I would be curious to see an explicit expression of the invariant functions ...
IMeasy's user avatar
  • 3,707
2 votes
0 answers
157 views

Linear projection from a point preserves flatness

Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
Chen's user avatar
  • 1,583
1 vote
1 answer
175 views

Subbundle generated by linearly dependent sections

On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
user avatar
0 votes
1 answer
76 views

How to calculate the camera 3D position if I know 4 points in the picture of the camera [closed]

I've got a picture of - let's say a - table and I measured 4 Points on this table. I define my table as "ground level", so my Points got the coordinates (px, py, 0). Is it possible to calculate the ...
tge's user avatar
  • 1
14 votes
1 answer
549 views

Projective-invariant differential operator

This question was originally asked on Math StackExchange. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = ...
user76284's user avatar
  • 1,793
7 votes
1 answer
634 views

A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof: Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
Đào Thanh Oai's user avatar
4 votes
1 answer
337 views

Is it a new method to construction of a conic, how can prove?

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows: Let $L_1, L_2$ be two parallel lines, let ...
Đào Thanh Oai's user avatar
2 votes
1 answer
349 views

Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$. Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...
Puzzled's user avatar
  • 8,832
2 votes
0 answers
99 views

Projectively flat Weyl connection on closed higher genus surface

A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel ...
user avatar
2 votes
0 answers
215 views

Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}. $$ This is related to a 4d SU(N) Yang-...
annie marie cœur's user avatar
15 votes
2 answers
729 views

Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

I've got ten (projective) planes in projective 3-space: \begin{align} &x=0\\ &z=0\\ &t=0\\ &x+y=0\\ &x-y=0\\ &z+t=0\\ &x-y-z=0\\ &x+y+z=0\\ &x-y+t=0\\ &x+y-t=0 ...
მამუკა ჯიბლაძე's user avatar
2 votes
0 answers
58 views

Is the finite projective plane stable as an extremal set system?

Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
GMB's user avatar
  • 1,379
3 votes
0 answers
84 views

Are there local invariants for smooth planes?

A smooth plane is a smooth double fibration $$ \mathbb{RP}^2 \overset{\pi_1}{\longleftarrow} PT\mathbb{RP}^2 \overset{\pi_2}{\longrightarrow} \mathbb{RP}^2 $$ where the system of curves $\pi_1(\pi_2^{...
alvarezpaiva's user avatar
  • 13.2k
1 vote
1 answer
104 views

Osculating spaces of intersection of two varieties

Let $Z = X\cap Y\subset\mathbb{C}^N$ be a manifold given as the intersection of two manifolds $X,Y$ intersecting transversally along $Z$. Let $T_p^kX,T_p^kY,T_p^kZ$ be the $k$-osculating spaces at $p\...
user avatar
2 votes
0 answers
94 views

Boundary at infinity for projective manifolds?

Given a compact real projective manifold, is there something "like" a boundary at infinity for its universal cover? In the case of compact projective manifolds obtained from divisible convex sets (...
alvarezpaiva's user avatar
  • 13.2k
22 votes
6 answers
2k views

About the definition of E8, and Rosenfeld's "Geometry of Lie groups"

I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
Pierre's user avatar
  • 2,145
5 votes
1 answer
238 views

Intersections in $\mathbb{P}^1\times\mathbb{P}^1$

Let $F$ be an algebraically closed field and $\mathbb{P}^1$ the projective line over $F$. Suppose $V_1, V_2$ are two 1-dimensional subvarieties of the 2-dimensional variety $\mathbb{P}^1\times\mathbb{...
Tsi's user avatar
  • 61
0 votes
0 answers
168 views

Derived Category of the Fano 4fold variety of lines

Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal ...
IMeasy's user avatar
  • 3,707
1 vote
0 answers
263 views

Dimension projectivised tangent space equal dimension variety+1

I'm reading some lecture notes (unfortunatey in italian, https://me.unitn.it/system/files/Bernardi%20Alessandra/tesi_teroni_1.pdf , page 5), where there's this statement (without proof): Suppose we ...
christmas_light's user avatar

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