\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

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

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \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{b}_{ 7 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 9 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \end{align*}
Basic information
dimension
9
index
10
Euler characteristic
10
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{9})$
$\mathrm{PGL}_{ 10 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{9})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{9})$
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.