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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 ... 159 views

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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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 $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+... 1 vote 0 answers 74 views ### Fitting point on a Quadric curve [closed] I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ... 1 vote 0 answers 45 views ### Dimensions of the intersection of 8 quadrics Suppose e_i,q_i \in \mathbb{R}^3, 1\leq i \leq 3 with \Vert e_i \Vert=1 are known. Define the projection on the plane orthogonal to e_i P_i= I-e_i e_i^T where I is the \mathbb{R}^{3\times ... 5 votes 1 answer 215 views ### Are quadrics the cones of maximal symmetry? A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural: One deep ... 10 votes 1 answer 452 views ### Fano Schemes of Intersections of Quadrics Let g\geqslant 2, and denote by \mathrm{X}=\mathrm{Q}_1\cap\mathrm{Q}_2\subset\mathbf{P}^{2g+1} a smooth intersection of quadrics. By considering the pencil generated by \mathrm{Q}_1,\mathrm{Q}_2... 1 vote 0 answers 48 views ### How explicitly write a projective transformation between the conics over the univariate function field? Consider the quadratic forms$$ Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2 $$over the rational function field \mathbb{F}_p(t), where p > 2 is a prime such that t^2 + 1 is ... 2 votes 1 answer 334 views ### Linear subspaces in quadric hypersurfaces Consider H_1,H_2,H_3\subset\mathbb{P}^{2m+1} three general linear subspaces of projective dimension m. Then there exists a quadric hypersurface Q^{2m}\subset\mathbb{P}^{2m+1} containing H_1,... 2 votes 1 answer 108 views ### Rational quadric bundles and group quotients Suppose I have a rational projective variety X and a quadric bundle Q \to X such that the total space of Q is rational. Assume now that I operate on X with a finite group G and that the ... 1 vote 0 answers 337 views ### A Special Case of Maximal Rank Conjecture A special case of maximal rank conjecture states that for a general curve C and general points p_1,\dots ,p_n\in C the map$$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$... 2 votes 1 answer 944 views ### Irreducible variety I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ... 3 votes 0 answers 339 views ### Rational normal curves on quadrics Given a quadric Q\subseteq\mathbb{P}^r and points p_1,\dots,p_{r+2}\in Q in linear general position, a naive dimension count suggests that one should expect finitely many rational normal curves ... 1 vote 0 answers 74 views ### Quadrics passing through a point of a variety that are parametrized by a quadric Let X\subset\mathbb{P}^{N} be a n-dimensional algebraic variety and let x\in X. Let us suppose that$$ \hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\} $$... 4 votes 2 answers 375 views ### Unfamiliar prime-generating polynomials related to Heegner numbers I just stumbled on a set of prime-generating polynomials of the form$$9 n^2-3 H n+H (H+1)/4$$(where H is a Heegner number >11), which generate the same number of distinct primes as their more ... 1 vote 1 answer 227 views ### Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane Consider a rational map u : \mathbb{CP}^1 \to \mathbb{CP}^4 of degree~d, such that the image lies in a fixed 3-dimensional quadric Q^3. In other words, its image is a rational curve in Q^3 \... 1 vote 0 answers 149 views ### Pascal and Brianchon's theorems generalized for hyperbolic paraboloid I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ... 3 votes 1 answer 636 views ### Quadrics and Moduli Spaces It is well known that \overline{M}_{0,5}, the moduli space of 5-pointed rational curves, can be realized as the blow-up of \mathbb{P}^2 in four general points. Therefore, we may interpret \... 1 vote 1 answer 850 views ### pencil of quadrics consisting of singular quadrics A pencil l of quadrics in \mathbb{P}^4_{<x_0,\cdots,x_4>} consists of singular quadrics only if: (1) quadrics in l have a common singular point; or (2) quadrics in l contain a common ... 4 votes 1 answer 678 views ### Automorphism group of a smooth quadric Q\subset\mathbb{P}^4 Let Q be the smooth quadric threefold in \mathbb{P}^4_{\mathbb{C}} defined by the equation x_0x_4+x_1x_3+x_2^2=0. Is it true that the automorphism group of Q is SO(Q;\mathbb{C}) which is ... 1 vote 1 answer 147 views ### Dimension of binary motives of a quadric Let Q be a anisotropic quadric of dimension d over k. We work in the category of effective Chow-Motives over k. Let T be the Tate-Motive. For a motive M we write M(l) for its l-th Tate-... 0 votes 1 answer 914 views ### Singular irreducible quadrics Let Q\subset\mathbb{P}^n be the quadric hypersurface defined by$$x_0^2+x_1^2+...+x_k^2 =0.$$If 2\leq k\leq n-1 then Q is irreducible and Sing(Q) is a linear space of dimension n-k-1. If ... 5 votes 2 answers 771 views ### Is the Lie quadric Q^3 isomorphic to the Lagrangian Grassmannian \operatorname{LG}(2,4)? \DeclareMathOperator\LG{LG}In the paper The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE (published on Proc. London Math. Soc.), I've found an interesting ... 4 votes 1 answer 625 views ### Blow-up of \mathbb{P}^4 along a quadric surface Let Q\subset\mathbb{P}^3\subset\mathbb{P}^4 be a smooth quadric surface, and let X = Bl_Q\mathbb{P}^4 the blow-up of \mathbb{P}^4 along Q. Let H be the pull-back of the hyperplane section of ... 6 votes 3 answers 1k views ### Automorphisms of a smooth quadric surface Q\subset\mathbb{P}^{3} Let Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3} be a smooth quadric surface. We have the following two actions on Q:$$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\... 1 vote
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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 ...
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 ...
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 \...