Questions tagged [projective-geometry]

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5
votes
1answer
101 views

Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric

Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric ...
6
votes
0answers
102 views

Picard group of resolution

Let $X$ be a normal variety and $f:Y\rightarrow X$ a birational morphism, contracting exceptional divisors $E_1,\dots,E_k$ onto the singular locus of $X$, with $Y$ smooth. In this situation is $Pic(Y)...
6
votes
0answers
113 views

Lefschetz type theorems for linear sections

Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has ...
1
vote
1answer
95 views

Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
3
votes
0answers
154 views

Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
1
vote
1answer
85 views

How to find the optimal lines?

Does anyone know anyway or any algorithm that can exactly and/or numerically find lines $\left\{ l_{i}\right\} _{i=1}^{n_{k}+2}$ that maximizes $$\min_{1\le i<j\le n_{k}+2}\text{angle}\left(l_{i},...
1
vote
0answers
68 views

Infinite dimensional smooth projective geometry

Are there two infinite dimensional (Banach or Hilbert) manifolds $(P,L)$ which satisfy the axioms of a smooth projective geometry desribed in this page: Smooth Projectove Geometry
4
votes
0answers
99 views

Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities. Assume that there is an isomorphism $f:...
4
votes
1answer
97 views

Terminal $\mathbb{Q}$-factorial divisorial contractions

Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$....
2
votes
0answers
42 views

Chow form of closure of product of affine varieties given the chow forms of their closurs

This question is about the connection between $\overline{X\times Y}$ and $\overline{X}$,$\overline{Y}$ where $X\subset \mathbb{A}^{n},\;Y\subset\mathbb{A}^{m}$ are affine varities over an ...
3
votes
1answer
148 views

Moduli space of hyperplane sections of a projective variety

Let $k$ be a field and let $V$ a finite-dimensional vector space over $k$. Assume that $X\subset \mathbb{P}(V)$ is a closed subvariety. Does there exist a proper flat morphism $Y\to \mathbb{P}(V^*)$ ...
2
votes
0answers
162 views

Bertini theorem for connectedness

Let $X$ be a geometrically irreducible, possibly singular projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ ...
2
votes
0answers
66 views

Birational contraction of toric vector bundle

Let $X$ be the toric vector bundle over $\mathbb{P}(1,1,1,2)$ with grading matrix $$ \left(\begin{array}{cccccc} 1 & 1 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 & ...
3
votes
1answer
283 views

Ideal of rational normal curve of degree $d$

Let $A$ consist of the columns of the $2\times (d+1)$ matrix $$A=\begin{pmatrix} d & d-1 & \cdots & 1&0\\ 0 & 1 & \cdots & d-1 &d \end{pmatrix}$$ Then consider the ...
6
votes
1answer
142 views

Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics? ...
3
votes
0answers
198 views

Understanding this example of projective geometry in Algebraic matroids

In this paper by Evans and Hrushovski: Projective planes in algebraically closed fields, they characterize projective planes in algebraically closed fields. These are coordinated by the skew-fields ...
1
vote
1answer
110 views

Singular locus of a linear system of hyperplane sections

Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
3
votes
0answers
100 views

A question on Okounkov bodies

Let $X$ be an irreducible $n$-dimensional projective variety, and $$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$ a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
4
votes
3answers
175 views

Finding ellipse-ellipse intersections in $\mathbb R^2$

The setting is as follows: we considers 2 disks embedded in $\mathbb R^3,$ and are interested in projecting one disk (either) onto the plane of the other other, and then compute their area of ...
2
votes
1answer
122 views

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
1answer
271 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
0answers
67 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
1answer
110 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
2answers
487 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 ...
3
votes
0answers
103 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
1answer
112 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
2answers
225 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
votes
0answers
109 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 ...
1
vote
1answer
104 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 ...
0
votes
0answers
36 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 ...
8
votes
1answer
155 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 ...
3
votes
0answers
76 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 ...
7
votes
1answer
413 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 ...
1
vote
1answer
161 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
0answers
75 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
votes
0answers
109 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 ...
1
vote
0answers
150 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 ...
0
votes
1answer
165 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 $X$? ...
7
votes
1answer
104 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 ...
1
vote
1answer
125 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 $n$. ...
2
votes
0answers
310 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
vote
0answers
116 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 ...
3
votes
1answer
139 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 ...
2
votes
0answers
211 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 ...
0
votes
1answer
27 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
1answer
330 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
votes
2answers
441 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 ...
4
votes
1answer
192 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 ...
3
votes
1answer
374 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 ...
3
votes
0answers
115 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 $C$?

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