Grassmannian.info

A periodic table of (generalised) Grassmannians.

Projective space $\mathbb{P}^{5}$

There exist other realisations of this Grassmannian:
Basic information
dimension
5
index
6
Euler characteristic
6
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{5})$
$\mathrm{PGL}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{5})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{5})$
35
Projective geometry
minimal embedding

$\mathbb{P}^{5}\hookrightarrow\mathbb{P}^{ 5 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 3 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 20 }$.

degree
1
Hilbert series
1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.