\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 $\mathbb{P}^{13}$

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 \\ \mathrm{b}_{ 11 } &= 1 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \\ \mathrm{b}_{ 14 } &= 1 \end{align*}
Basic information
dimension
13
index
14
Euler characteristic
14
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 11 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$, $\mathrm{b}_{ 14 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{13})$
$\mathrm{PGL}_{ 14 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{13})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{13})$
195
Projective geometry
minimal embedding

$\mathbb{P}^{13}\hookrightarrow\mathbb{P}^{ 13 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 7 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 104 }$.

degree
1
Hilbert series
1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400600, 20058300, 37442160, 67863915, 119759850, 206253075, 347373600, ...
Exceptional collections
  • Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.