Questions tagged [quadrics]

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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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1answer
192 views

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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1answer
99 views

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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235 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 $$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$ ...
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1answer
762 views

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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253 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 ...
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69 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 $$ \hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\} $$ ...
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2answers
325 views

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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1answer
178 views

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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137 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 ...
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1answer
456 views

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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1answer
637 views

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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1answer
354 views

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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1answer
132 views

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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1answer
501 views

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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1answer
454 views

Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $LG(2,4)$?

On the paper http://arxiv.org/abs/1009.1364 (published on Proc. London Math. Soc.) I've found an interesting statement: The Lie quadric $Q^3$, i.e., the space of all points, lines and circles ...
5
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1answer
375 views

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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3answers
755 views

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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371 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 ...
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1answer
335 views

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 ...
3
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1answer
405 views

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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3answers
488 views

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 \...