Questions tagged [projective-geometry]
The projective-geometry tag has no usage guidance.
614
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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 ...
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 ...
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 ...
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 $\mathbb{P}^1(k)$, ...
4
votes
2
answers
602
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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 ...
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?
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\...
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 ...
3
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0
answers
125
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$\left< 15\right>^7/15$-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 ...
1
vote
1
answer
157
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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 ...
0
votes
0
answers
45
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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 ...
12
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1
answer
305
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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 ...
3
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0
answers
104
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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 ...
7
votes
1
answer
462
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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 ...
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 $X$ ...
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 ...
2
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0
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240
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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 ...
1
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0
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152
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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 ...
0
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1
answer
176
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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 $X$? ...
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 ...
3
votes
1
answer
228
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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
$n$. ...
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 ...
1
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0
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125
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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 ...
3
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1
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153
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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 ...
2
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0
answers
361
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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 ...
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 ...
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.
7
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2
answers
2k
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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 ...
3
votes
1
answer
410
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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 ...
2
votes
1
answer
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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 ...
2
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0
answers
139
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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 $C$?
3
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0
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151
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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 ...
2
votes
0
answers
157
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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 ...
1
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1
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175
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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 ...
0
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1
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76
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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 ...
14
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1
answer
549
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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) = ...
7
votes
1
answer
634
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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 ...
4
votes
1
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337
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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 ...
2
votes
1
answer
349
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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,...
2
votes
0
answers
99
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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 ...
2
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0
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215
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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-...
15
votes
2
answers
729
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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
...
2
votes
0
answers
58
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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 ...
3
votes
0
answers
84
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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^{...
1
vote
1
answer
104
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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\...
2
votes
0
answers
94
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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 (...
22
votes
6
answers
2k
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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 ...
5
votes
1
answer
238
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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{...
0
votes
0
answers
168
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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 ...
1
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0
answers
263
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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 ...