The quadrics tag has no wiki summary.

**-2**

votes

**1**answer

120 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 ...

**1**

vote

**1**answer

228 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 ...

**2**

votes

**1**answer

195 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 ...

**4**

votes

**3**answers

246 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,\; ...

**1**

vote

**0**answers

174 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 ...

**3**

votes

**1**answer

270 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**

votes

**1**answer

294 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 ...

**2**

votes

**2**answers

337 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 ...