Questions tagged [projective-geometry]
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615
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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, ...
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Matrix powers up to multiplicative factor
Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and
$$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$
I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \...
3
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0
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151
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Blowing-up a non reduced fiber
Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$.
I expect $\...
3
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0
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65
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Field automorphisms of projective spaces without the axiom of choice
Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
1
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1
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85
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Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$
Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...
2
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0
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87
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Nonlinear automorphisms of projective spaces and the axiom of choice
Let $k$ be a field and $\mathbf{P}$ a projective space over $k$. If we accept the axiom of choice (AC), then $\mathbf{P}$ has a basis and a dimension $m$, and if $m$ is finite, the automorphism group ...
2
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0
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93
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$0$-dimensional intersection in weighted projective space
Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by
$$
\phi(...
1
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1
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218
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Surfaces with rational double points
Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and ...
3
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1
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169
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Singularities of surfaces fibered in rational curves
Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point.
...
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Projectivization in the derived category of coherent sheaves
Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...
6
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485
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Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines
The simplest case of the Fundamental Theorem of Projective Geometry states that, if $f: \mathbb{R}^2\to\mathbb{R}^2$ is a bijection that preserves lines – in the sense that if $L\subseteq\mathbb{R}^2$ ...
3
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0
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37
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Anti-flag transitive affine planes
Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite.
Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...
2
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1
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181
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Question regarding linear system of projective space
I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2....
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1
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154
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Number of orbits for abelian group actions
Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
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0
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Is the ideal of the Veronese variety $V_{d,n}$ generated by quadrics?
Maybe it sounds like a silly question to the experts but I'm not able to find a proper reference in the web. Anyone knows if the ideal $I_{d,n}$ of the Veronese variety $V_{d,n}$ is generated by ...
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Anti-flag transitive projective planes
Let $\Gamma$ be an axiomatic projective plane, and suppose its automorphism group acts transitively on the anti-flags (the point-line pairs $(u,V)$ such that $u$ is not incident with $V$).
In the ...
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1
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162
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A variation on the projective Nullstellensatz
Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ for which $V(f_1,\dots, f_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ ...
6
votes
1
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690
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Singular curves of genus 1
Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can ...
4
votes
1
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218
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Volume of conic bundles
Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
4
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1
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222
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Contact variety to projective variety is equidimensional
I first asked this question at math.stackexchange with no success, so I decided to repost it here.
I am reading the paper "Weakly Defective Varieties" by L. Chiantini and C. Ciliberto, ...
3
votes
1
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Smooth surfaces in positive characteristic
Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form
$$
S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...
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Hypersurface on $\mathbb{P}^m\times\mathbb{P}^n$ [closed]
I have found a statement (Harris "Algebraic Geometry", p.269) saying that a hypersurface in $\mathbb{P}^m\times\mathbb{P}^n$ can be written as a single equation. I couldn't find the proof.
...
4
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1
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290
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Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
1
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0
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67
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Projective equivalence for quadrics and their image
Let $\mathbb{K}$ be the complex or real field and $(S_n,\mathcal{K})$ a $\mathbb{K}-$projective space of dimension $n\in\mathbb{N}^*$, where $\mathcal{K}$ is the collection of bijections $\kappa: S\to\...
6
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2
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674
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Smooth complete intersections
Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...
1
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1
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162
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Space of rational conics
Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.
Conisider the ...
1
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0
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35
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Image points to a plane and computing the covariance for a noisy observer
Let's assume a camera in space and an image point in this camera:
$t \in \mathbb{R}^3$ is the position of the camera in space.
$R \in \mathbb{R}^{3 \times 3}$ is the orientation of the camera in ...
5
votes
1
answer
269
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Mirror symmetry for K3 fibered Calabi-Yau threefolds
By a K3 fibered Calabi-Yau threefold, I mean a smooth projective threefold $X$ with trivial canonical class and
$h^{1,0}(X) =h^{2,0}(X) = 0$ that has a fibration $X \rightarrow \mathbb P^1$ whose ...
2
votes
1
answer
149
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Linear subspace in quadric hypersurfaces over a field
Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$.
Suppose also that $Q$ has a $K$-point and so $Q$ is ...
4
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0
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264
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How many ways are there to characterise $\mathbb{P}^n$?
Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
5
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1
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497
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Lines on quadric surfaces
Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
1
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2
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Embedding of a blow-up
In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H_1,H_2$ of type $(2,1)$ and set $Y = X\cap H_1\cap H_2$.
Let $Z$ be the blow-up of $X$ ...
5
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2
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548
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Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
3
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0
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258
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Clarifications involving automorphisms of projective planes and lines?
I have been learning some classical projective geometry recently and I am hoping to gain some clarity regarding various different automorphism groups. There are three different levels of generality ...
3
votes
1
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168
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Embedding quadric bundles
Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$.
Does there always ...
2
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0
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74
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Which projective planes are connected manifolds?
Inspired by a nice recent MO question, I thought I would ask a similar one: which projective planes $P$ can be given a topology and structure of a smooth connected manifold where the lines form smooth ...
9
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1
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467
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p-adic analogue of octonions
There are the complex p-adic numbers.
But what is the p-adic analogue of the Cayley–Dickson construction?
Or more important: What is the p-adic analogue of the octonions?
It would be nice if the (unit)...
0
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0
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Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$
This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...
2
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1
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221
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Equivalence of sequences of blowups of $\mathbb{P}^3$
Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
2
votes
1
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157
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Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?
First time, I found a line associated with antipodal points, detail:
Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
1
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0
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67
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Generalized polynomials and a $4$-point cross-ratio function on the non-degenerate complex $4$-quadric
The Grassmannian $Gr_1(\mathbb{C}^2)$ is another name for $\mathbb{P}^1$. If one endows $\mathbb{C}^2$ with a complex symplectic form, or if one prefers (since this will allow us to generalize in ...
4
votes
1
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545
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Cohomology of divisors on Hirzebruch surfaces
Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...
1
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0
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196
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Mori cone of Picard rank two varieties
Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...
2
votes
1
answer
408
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Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
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votes
1
answer
242
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Coefficients of elliptic curves over function fields
Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$.
Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...
2
votes
1
answer
232
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Blow-ups of surfaces over a field
Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure.
Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
5
votes
2
answers
498
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Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
2
votes
1
answer
195
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Trivial rational solution of a system of hyperplanes
Let us consider a vector space $ V $ over $ \mathbb{Q} $ of dim $6$. We denote all the two dimensional subspace in $ V $ by $ G(2,6) $ (The Grassmanian variety). One can define a map $ p $ from $ G(2,...
6
votes
2
answers
380
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Nef divisors on surfaces
Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:
$...
0
votes
1
answer
173
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Curves in conic bundles
Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...