Questions tagged [quadrics]
The quadrics tag has no usage guidance.
13
questions with no upvoted or accepted answers
6
votes
0
answers
196
views
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 ...
4
votes
0
answers
58
views
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
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...
3
votes
0
answers
351
views
Rational normal curves on quadrics
Given a quadric $Q\subseteq\mathbb{P}^r$ and points $p_1,\dots,p_{r+2}\in Q$ in linear general position, a naive dimension count suggests that one should expect finitely many rational normal curves ...
1
vote
0
answers
151
views
Intersection of three quadrics: associating something geometric to these analogously to intersection of two quadrics
Consider the smooth intersection of two $4$-dimensional quadrics $Y = Q \cap Q' \subset P^5$. To the Fano threefold $Y$ we can associate a genus $2$ curve as follows. Take the pencil of quadrics $\{ ...
1
vote
0
answers
68
views
Linear subspaces of quadric surface over arbitrary fields
I was wondering if there already exists some text that present classical results on quadrics in a modern language. Ideally using the language of schemes, however the setting of varieties over ...
1
vote
0
answers
68
views
Projective equivalence for quadrics and their image
Let $\mathbb{K}$ be the complex or real field and $(S_n,\mathcal{K})$ a $\mathbb{K}-$projective space of dimension $n\in\mathbb{N}^*$, where $\mathcal{K}$ is the collection of bijections $\kappa: S\to\...
1
vote
0
answers
46
views
Dimensions of the intersection of 8 quadrics
Suppose $e_i,q_i \in \mathbb{R}^3$, $1\leq i \leq 3$ with $\Vert e_i \Vert=1$ are known.
Define the projection on the plane orthogonal to $e_i$
$P_i= I-e_i e_i^T$ where $I$ is the $\mathbb{R}^{3\times ...
1
vote
0
answers
48
views
How explicitly write a projective transformation between the conics over the univariate function field?
Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
1
vote
0
answers
347
views
A Special Case of Maximal Rank Conjecture
A special case of maximal rank conjecture states that for a general curve $C$ and general points $p_1,\dots ,p_n\in C$ the map
$$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$
...
1
vote
0
answers
74
views
Quadrics passing through a point of a variety that are parametrized by a quadric
Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that
$$
\hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\}
$$
...
1
vote
0
answers
151
views
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 ...
1
vote
0
answers
428
views
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 ...
0
votes
0
answers
86
views
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 ...