Grassmannian.info

A periodic table of (generalised) Grassmannians.

Quadric $\mathrm{Q}^{10}$

Basic information
dimension
10
index
10
Euler characteristic
12
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 11 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{10})$
$\mathrm{PSO}_{ 12 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{10})$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{10})$
66
Projective geometry
minimal embedding

$\mathrm{Q}^{10}\hookrightarrow\mathbb{P}^{ 11 }$

degree
2
Hilbert series
1, 12, 77, 352, 1287, 4004, 11011, 27456, 63206, 136136, 277134, 537472, 999362, 1790712, 3105322, 5230016, 8580495, 13748020, 21559395, 33153120, ...
Exceptional collections
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.