# Questions tagged [quadrics]

The quadrics tag has no usage guidance.

36
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### Intersection of three quadrics: associating something geometric to these analogously to intersection of two quadrics

Consider the smooth intersection of two $4$-dimensional quadrics $Y = Q \cap Q' \subset P^5$. To the Fano threefold $Y$ we can associate a genus $2$ curve as follows. Take the pencil of quadrics $\{ ...

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### Entanglement, quadrics and $\mathbb{P}^2(\mathbb{C}^3)$ [closed]

First of all: I apologise in advance for if my question will be arid, wrong written or even nonsensical.
I was at a talking with a professor last week, and the question of "Entanglement and ...

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### Quadric contain tangent variety of a curve in $\mathbb{P}^5$

Let $Q^4 \subset \mathbb{P}^5$ a smooth quadric over $\mathbb{C}$
which is via Pluecker map isomorphic
to Grassmannian of lines $\mathbb{G}(1,\mathbb{P}^3)$
in $\mathbb{P}^3$.
Consider following ...

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### Is there any way of explaining the Cayley/Beltrami–Klein metric to undergrads?

How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a ...

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### Linear subspaces of quadric surface over arbitrary fields

I was wondering if there already exists some text that present classical results on quadrics in a modern language. Ideally using the language of schemes, however the setting of varieties over ...

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### Number of points of a quadric hypersurface over a finite field

Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...

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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\...

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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 ...

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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 ...

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### Fundamental group of the complement of some quadric cones

cross-posting from MathSE
Problem
Consider the domain
$$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$
and the map
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...

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### Fitting point on a Quadric curve [closed]

I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...

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### Dimensions of the intersection of 8 quadrics

Suppose $e_i,q_i \in \mathbb{R}^3$, $1\leq i \leq 3$ with $\Vert e_i \Vert=1$ are known.
Define the projection on the plane orthogonal to $e_i$
$P_i= I-e_i e_i^T$ where $I$ is the $\mathbb{R}^{3\times ...

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1
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### Are quadrics the cones of maximal symmetry?

A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural:
One deep ...

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### Fano Schemes of Intersections of Quadrics

Let $g\geqslant 2$, and denote by $\mathrm{X}=\mathrm{Q}_1\cap\mathrm{Q}_2\subset\mathbf{P}^{2g+1}$ a smooth intersection of quadrics. By considering the pencil generated by $\mathrm{Q}_1,\mathrm{Q}_2$...

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### How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...

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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,...

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votes

1
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### Rational quadric bundles and group quotients

Suppose I have a rational projective variety $X$ and a quadric bundle $Q \to X$ such that the total space of $Q$ is rational. Assume now that I operate on $X$ with a finite group $G$ and that the ...

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### A Special Case of Maximal Rank Conjecture

A special case of maximal rank conjecture states that for a general curve $C$ and general points $p_1,\dots ,p_n\in C$ the map
$$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$
...

2
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answer

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### Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...

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### Rational normal curves on quadrics

Given a quadric $Q\subseteq\mathbb{P}^r$ and points $p_1,\dots,p_{r+2}\in Q$ in linear general position, a naive dimension count suggests that one should expect finitely many rational normal curves ...

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### Quadrics passing through a point of a variety that are parametrized by a quadric

Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that
$$
\hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\}
$$
...

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votes

2
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### Unfamiliar prime-generating polynomials related to Heegner numbers

I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more ...

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### Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane

Consider a rational map $u : \mathbb{CP}^1 \to \mathbb{CP}^4$ of degree~$d$, such that the image lies in a fixed 3-dimensional quadric $Q^3$. In other words, its image is a rational curve in $Q^3 \...

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### Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...

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### Quadrics and Moduli Spaces

It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret $\...

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### pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if:
(1) quadrics in $l$ have a common singular point; or
(2) quadrics in $l$ contain a common ...

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### Automorphism group of a smooth quadric $Q\subset\mathbb{P}^4$

Let $Q$ be the smooth quadric threefold in $\mathbb{P}^4_{\mathbb{C}}$ defined by the equation
$x_0x_4+x_1x_3+x_2^2=0$.
Is it true that the automorphism group of $Q$ is $SO(Q;\mathbb{C})$ which is ...

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1
answer

147
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### Dimension of binary motives of a quadric

Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th Tate-...

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### Singular irreducible quadrics

Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x_0^2+x_1^2+...+x_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.
If $...

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### Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $\operatorname{LG}(2,4)$?

$\DeclareMathOperator\LG{LG}$In the paper The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE (published on Proc. London Math. Soc.), I've found an interesting ...

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### Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...

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### Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\...

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### Odd-Dimensional Complex Quadrics

It's well known that the even-dimensional complex quadric $Q_{2n}$, defined by the equation $z_1^2+\cdots +z_{2n+2}^2=0$ in complex projective space $CP^{2n+1}$, is diffeomorphic with the Grassmann ...

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### Sections of a fibration in intersections of quadrics

Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces ...

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### degeneration of quadric bundles

Suppose I have a smooth 2-dimensional quadric bundle $f:X\to S$ over a surface $S$. Suppose furthermore that the discriminant locus $\Delta \subset S$ is smooth. Can I immedately conclude that the ...

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### quadrics containing the tangential variety of a curve

Let $C\subset\mathbb{P}^r$ be a smooth nondegenerate curve (not contained in any hyperplane) of degree $d$ genus $g>0$. Consider the tangential variety $X$ of $C$: $X=\cup_{p\in C}T_pC\subset \...