# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Quadric $\mathrm{Q}^{6}$

There exist other realisations of this Grassmannian:
Basic information
dimension
6
index
6
Euler characteristic
8
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{6})$
$\mathrm{PSO}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{6})$
$\mathbb{Z}/3\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{6})$
28
Projective geometry
minimal embedding

$\mathrm{Q}^{6}\hookrightarrow\mathbb{P}^{ 7 }$

degree
2
Hilbert series
1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, 311696, ...
Exceptional collections
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{Q}^{6})$ are given by:

Homological projective duality