Questions tagged [projective-geometry]
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614
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Maximal abelian subgroups of the full collineation group $\mathrm{P\Gamma L}_3(q)$
Is there a convenient list of the maximal abelian subgroups of the projective semilinear group $\mathrm{P\Gamma L}_3(K) \cong \mathrm{PGL}_3(K) \rtimes \mathrm{Gal}(K)$ for $K$ a finite field?
This is ...
2
votes
0
answers
183
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Divisorial contraction to a non-normal variety
Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
1
vote
0
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126
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Trisecant lines to curves in $\mathbb{P}^3$: a reference request
I'm starting to study the geometry of the locus of trisecant lines to a space curve $C \subset \mathbb{P}^3$. The main references, cited in almost all the papers, regarding the specific calculations ...
5
votes
2
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246
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Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$
Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$.
If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...
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0
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141
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Cubic surface in $\mathbb{P}^3$ singular along a line
Maybe it is a stupid question but I'm not able to find the answer anywhere else.
My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
0
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0
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92
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Lines through the origin every pair of which meet at the same angle
This item isn't getting attention, so I'll try it here:
begin quote
The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles.
The three lines ...
1
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1
answer
222
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Smoothness of moduli spaces of stable maps
If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}_{...
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0
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142
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Kähler fiber space with base and fiber projective
Let $X$ be a Kähler manifold, $Y$ be a projective manifold, if $X$ exits a smooth fibration over $Y$ such that all the fibers are projective manifolds, then is $X$ a projective mannifold?
If we do not ...
0
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1
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248
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Does projective transformation preserve convexity? [closed]
Does projective transformation preserve convexity?
Notice: Ignore the trivial case which projects a convex curve to a straight line.
5
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1
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278
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Isomorphisms of complete intersections
Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
3
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78
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Infinite-dimensional quasifields
In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and ...
3
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163
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Divisorial contractions and singularities
I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...
2
votes
1
answer
528
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A question on a Macaulay2 computation
I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
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1
answer
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What is the automorphism group of the projective line minus $n$ points?
$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...
2
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572
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Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$
Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
7
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1
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424
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Strict transform of a tangent curve under blow-up
$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
4
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0
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213
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Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
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532
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When a fibration over a projective manifold is projective?
Let $X$ be a complex manifold, $B$ be a complex projective manifold, consider a smooth fibration $\pi:X\rightarrow B$ such that all the fibers of $\pi$ are simply-connected projective manifolds, then ...
2
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171
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Understanding why $\pi_{3}(N) = \mathbb{Z}_{(2n, n^2)}$?
I am trying to understand the paper Arkowitz and Golasinski - Co-$H$ structures on Moore spaces of type $(G, 2)$:
In section 4, titled "Homotopy elements of finite order", the authors say $\...
3
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1
answer
192
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Vanishing locus generic section $(\mathrm{sym}^2 \mathcal{R})(1)$
Let $n = 2m$ be an even integer and let $\mathcal{R}$ the tautological bundle on the Grassmannian $\mathrm{Gr}(2,n)$. I am looking for an explicit description of the degener
The bundle $(\mathrm{Sym}^...
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60
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Does base point free linear system of polynomials generate higher degree polynomials
Let $k$ be an algebraically closed field. Let $S=k[x_0,\ldots, x_4]$ be the ring of polynomials. We set $S^i$ to the graded piece of degree $i$ polynomials. Let $H$ be a hyperplane of $S^5$ with no ...
2
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194
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2 K3s and cubic fourfolds containing a plane
Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...
13
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2
answers
558
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A geometric definition of the addition law on abelian surfaces
Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.
Is there a ...
6
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1
answer
282
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Singular cardinal $\kappa$ with projective plane such that all edges have cardinality $<\kappa$
Is there an infinite singular cardinal $\kappa$ such that there is a set $E\subseteq{\cal P}(\kappa)$ with the following properties?
$|e| < \kappa$ for all $e\in E$,
whenever $\alpha\neq\beta\in \...
0
votes
1
answer
45
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Non-pencil infinite projective plane with edges of different cardinalities
A projective plane is a hypergraph $H=(V,E)$ such that
if $e_1\neq e_2 \in E$ then $|e_1\cap e_2| = 1$, and
for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$.
Is there a projective ...
4
votes
0
answers
114
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Klein geometry associated to a degenerate conic
In his study of non-euclidean geometries,
Felix Klein considers the group of projective transformations
acting on the real projective plane whose extensions
to the complex projective plane preserve a ...
2
votes
1
answer
174
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Decomposition of a morphism with positive dimensional fibers
It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a ...
3
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Presentations of $\mathbf{PGL}_3(\mathbb{F}_q)$ by three involutions
If I am not mistaken, the group $\mathbf{PGL}_3(\mathbb{F}_q)$, with $\mathbb{F}_q$ the finite field with $q$ elements, can be generated by three involutions.
Where could I find such representations ?
...
1
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1
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Presentations of $\mathbf{𝐏𝐆𝐋}_3(\mathbb{F}_2)$ by three involutions, 2
I am searching for a presentation of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
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0
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170
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Corollary of Mori’s theorem
Is there a direct proof of the corollary of Mori's theorem which says: If a projective complex manifold does not contain a rational curve then $ K_X $ is nef
5
votes
2
answers
381
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Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions
I am searching for (two) presentations of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
1
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0
answers
231
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Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$
Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar.
Define an $\mathrm{SL}(2)$-action on $\...
9
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1
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520
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Degree of secant varieties of Veronese varieties
Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.
Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
9
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0
answers
871
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A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?
In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $...
4
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1
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252
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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
...
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1
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207
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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
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0
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173
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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
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1
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113
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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},...
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0
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92
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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
3
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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:...
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1
answer
192
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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
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0
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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
1
answer
323
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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 $Y\to \mathbb{P}(V^*)$ ...
3
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1
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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 $X$ ...
4
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1
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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
1
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283
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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?
...
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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 ...
2
votes
1
answer
290
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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 $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
2
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0
answers
160
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A question on Okounkov bodies
Let $X$ be an irreducible $n$-dimensional projective variety, and
$$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
3
answers
476
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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 ...