# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{9}$

There exist other realisations of this Grassmannian:
Basic information
dimension
9
index
10
Euler characteristic
10
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{9})$
$\mathrm{PGL}_{ 10 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{9})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{9})$
99
Projective geometry
minimal embedding

$\mathbb{P}^{9}\hookrightarrow\mathbb{P}^{ 9 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 5 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 54 }$.

degree
1
Hilbert series
1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• 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 $\mathrm{c}_1(\mathbb{P}^{9})$ are given by:

Homological projective duality